# From phonemes to images: levels of representation in a recurrent neural
model of visually-grounded language learning

Lieke Gelderloos
Tilburg University
<EMAIL_ADDRESS>&Grzegorz Chrupała
Tilburg University
<EMAIL_ADDRESS>
###### Abstract

We present a model of visually-grounded language learning based on stacked
gated recurrent neural networks which learns to predict visual features given
an image description in the form of a sequence of phonemes. The learning task
resembles that faced by human language learners who need to discover both
structure and meaning from noisy and ambiguous data across modalities. We show
that our model indeed learns to predict features of the visual context given
phonetically transcribed image descriptions, and show that it represents
linguistic information in a hierarchy of levels: lower layers in the stack are
comparatively more sensitive to form, whereas higher layers are more sensitive
to meaning.

00footnotetext:  This work is licensed under a Creative Commons Attribution
4.0 International License. License details:
http://creativecommons.org/licenses/by/4.0/

##  1 Introduction

Children acquire their native language with little and weak supervision,
exploiting noisy correlations between speech, visual, and other sensory
signals, as well as via feedback from interaction with their peers and
parents. Understanding this process is an important scientific challenge in
its own right, but it also has potential to generate insights useful in
engineering efforts to design conversational agents or robots. Computationally
modeling the ability to learn linguistic form–meaning pairings has been the
focus of much research, under scenarios simplified in a variety of ways, for
example:

  * •

distributional learning from pure word-word co-occurrences with no perceptual
grounding [Landauer et al., 1998, Kiros et al., 2015];

  * •

cross-situational learning with word sequences and sets of symbols
representing sensory input [Siskind, 1996, Fazly et al., 2010];

  * •

cross-situational learning using sensory audio and visual input, but with
extremely limited sets of words and objects [Roy and Pentland, 2002, Iwahashi,
2003].

Some recent models have used more naturalistic, larger-scale inputs, for
example in cross-modal distributional semantics [Lazaridou et al., 2015] or in
implementations of the acquisition process trained on images paired with their
descriptions [Chrupała et al., 2015]. While in these works the representation
of the visual scene consists of pixel-level perceptual data, the linguistic
input consists of sentences segmented into discrete word symbols. In this
paper we take a step towards addressing this major limitation, by using the
phonetic transcription of input utterances. While this type of input is
symbolic rather than perceptual, it goes a long way toward making the setting
more naturalistic, and the acquisition problem more challenging: the learner
may need to discover structure corresponding to morphemes, words and phrases
in an unsegmented string of phonemes, and the length of the dependencies that
need to be detected grows substantially when compared to word-level models.

#### Our contributions

We design and implement a simple architecture based on stacked recurrent
neural networks with Gated Recurrent Units [Chung et al., 2014]: our model
processes the utterance phoneme by phoneme while building a distributed low-
dimensional semantic representation through a series of recurrent layers. The
model learns by projecting its sentence representation to image space and
comparing it to the features of the corresponding visual scene.

We train this model on a phonetically transcribed version of MS-COCO [Lin et
al., 2014] and show that it is able to successfully learn to understand
aspects of sentence meaning from the visual signal, and exploits temporal
structure in the input. In a number of experiments we show that different
levels in the stack of recurrent layers represent different aspects of
linguistic structure. Low levels focus on local, short-time-scale spans of the
input sequence, and are comparatively more sensitive to form. The top level
encodes global aspects of the input sequence and is sensitive to visually
salient elements of its meaning.

##  2 Related work

A major part of learning language consists in learning its structure, but in
order to be able to communicate it is also of crucial importance to learn the
relation of words to entities in the world. Grounded lexical acquisition is
often modeled as cross-situational learning, a process of rule-based [Siskind,
1996] or statistical [Fazly et al., 2010, Frank et al., 2007] inference of
word-to-referent mappings. Cross-situational models typically work on word-
level language input and symbolic representations of the context. However,
infants have to learn from continuous perceptual input. ?) partially remedy
this shortcoming and propose a model of learning word meanings from text
paired with continuous image representations; the limitation of their work is
the toy evaluation dataset.

Recent experimental and computational studies have found that co-occurring
visual information may help to learn word forms [Thiessen, 2010, Cunillera et
al., 2010, Glicksohn and Cohen, 2013, Yurovsky et al., 2012]. This suggests
that acquisition of word form and meaning are interactive, rather than
separate.

The Cross-channel Early Lexical Learning (CELL) model of ?) and the more
recent work of ?) take into account the continuous nature of the speech
signal, and incorporate visual information as well. The CELL model learns to
discover words in continuous speech through co-occurence with their visual
referent, but the visual input only consists of the shape of single objects,
effectively bypassing referential uncertainty. ?) propose a probabilistic
joint model of word segmentation and meaning acquisition from raw speech and a
set of possible referents that appear in the context. In both ?) and ?) the
visual context is considerably less noisy and ambiguous than that available to
children.

There is an extensive line of research on image captioning (see ?) for a
recent overview). Typically, captioning models learn to recognize high-level
image features and associate them with words. Inspired by both image
captioning research and cross-situational human language acquisition, two
recent automatic speech recognition models learn to recognize word forms from
visual data. In ?), language input consists of single spoken words and visual
data consists of image fragments, which the model learns to associate. ?)
employ two convolutional neural networks, a visual object recognition model
and a word recognition model, and an embedding alignment model that learns to
map recognized words and objects into the same high-dimensional space.
Although the object recognition works on the raw visual input, the speech
signal is segmented into words before presenting it to the word recognition
model. As both ?) and ?) work with word-sized chunks of speech, they bypass
the segmentation problem that human language learners face.

Character-level input representations have recently gained attention in NLP.
?) and ?) use bidirectional LSTMs to compose characters into word embeddings,
while ?) propose machine translation model with character level output. These
approaches exploit character-level information but crucially they assume that
word boundaries are available in the input.

Character-level neural NLP without explicit word boundaries in the input is
studied in cases where fixed vocabularies are inherently problematic, e.g.
with combined natural and programming language input [Chrupała, 2013] or when
specifically dealing with misspelled words in automatic writing feedback [Xie
et al., 2016].

Character-level language models are analyzed in ?) and ?). Both studies show
that character-level recurrent neural networks are sensitive to long-range
dependencies: for example by keeping track of opening and closing parentheses
over stretches of text. ?) describe the hierarchical organization that emerges
during training, with higher layers processing information over longer
timescales. In our work we show related effects in a model of visually-
grounded language learning from unsegmented phonemic strings.

We use phonetic transcription of full sentences as a first step towards large-
scale multimodal language learning from speech co-occurring with visual
scenes. The use of phonetic transcription rather than raw speech signal
simplifies learning and allows us to perform experiments on the encoding of
linguistic knowledge as reported in section 4 without additional annotation.
Our goal is to model a multi-modal language learning process that includes the
segmentation problem faced by human language learners. In contrast to
character-level NLP and language modeling approaches, our input data therefore
does not contain word boundaries or strong cues such as whitespace and
punctuation.

In contrast to ?) and ?), the visual input to our model consists of high-level
visual features, which means it contains ambiguity and noise. In contrast to
?) and ?), we consider full utterances rather than separate words.

To our knowledge, there is no work yet on multimodal phoneme or character-
level language modeling with visual input. Although the language input in this
study is low-level-symbolic rather than perceptual, the learning problem is
similar to that of a human language learner: discover language structure as
well as meaning, based on ambiguous and noisy data from another modality.

?) simulate visually grounded human language learning in face of noise and
ambiguity in the visual domain. Their model predicts visual context given a
sequence of words. While the visual input consists of a continuous
representation, the language input consists of a sequence of words. The aim of
this study is to take their approach one step further towards multimodal
language learning from raw perceptual input. ?) develop techniques for
understanding and interpretation of the representations of linguistic form and
meaning in recurrent neural networks, and apply these to word-level models. In
our work we share the goal of revealing the nature of emerging
representations, but we do not assume words as their basic unit. Also, we are
especially concerned with the emergence of a hierarchy of levels of
representations in stacked recurrent networks.

##  3 Models

Consider a learner who sees a person pointing at a scene and uttering the
unfamiliar phrase Look, the monkeys're playing. We may suppose that the
learner will update her language understanding model such that the subsequent
utterance of this phrase will evoke in her mind something close to the
impression of this visual scene. Our model is a particular instantiation of
this simple idea.

###  3.1 Phon GRU

The architecture of our main model of interest, Phon GRU is schematically
depicted in Figure 2 and consists of a phoneme encoding layer, followed by a
stack of $K$ Gated Recurrent Neural nets, followed by a densely connected
layer which maps the last hidden state of the top recurrent layer to a vector
of visual features.

Figure 1: A three-timestep slice of the stacked recurrent architecture with
three hidden layers.

---
A young woman riding a horse holding a flag

Figure 2: (Top) Example of a postprocessed phonetic transcription from eSpeak
used as input to the Phon GRU model. (Bottom) Corresponding image.

Gated Recurrent Units (GRU) were introduced in ?) and ?) as an attempt to
alleviate the problem of vanishing gradient in standard simple recurrent nets
as known since the work of ?). GRUs have a linear shortcut through timesteps
which bypasses the nonlinearity and thus promotes gradient flow. Specifically,
a GRU computes the hidden state at current time step, $\mathbf{h}_{t}$, as the
linear combination of previous activation $\mathbf{h_{t-1}}$, and a new
candidate activation $\mathbf{\tilde{h}}_{t}$:

$\mathrm{gru}(\mathbf{x}_{t},\mathbf{h}_{t-1})=(1-\mathbf{z}_{t})\odot\mathbf{h}_{t-1}+\mathbf{z}_{t}\odot\mathbf{\tilde{h}}_{t}\vspace{-.1cm}$
(1)

where $\odot$ is elementwise multiplication, and the update gate activation
$\mathbf{z_{t}}$ determines the amount of new information mixed in the current
state:

$\mathbf{z}_{t}=\sigma_{s}(\mathbf{W}_{z}\mathbf{x}_{t}+\mathbf{U}_{z}\mathbf{h}_{t-1})$
(2)

The candidate activation is computed as:

$\mathbf{\tilde{h}}_{t}=\sigma(\mathbf{W}\mathbf{x}_{t}+\mathbf{U}(\mathbf{r}_{t}\odot\mathbf{h}_{t-1}))$
(3)

The reset gate $\mathbf{r_{t}}$ determines how much of the current input
$\mathbf{x_{t}}$ is mixed in the previous state $\mathbf{h}_{t-1}$ to form the
candidate activation:

$\mathbf{r}_{t}=\sigma_{s}(\mathbf{W}_{r}\mathbf{x}_{t}+\mathbf{U}_{r}\mathbf{h}_{t-1})$
(4)

By applying the $\mathrm{gru}$ function repeatedly a GRU layer maps a sequence
of inputs to a sequence of states:

$\mathrm{GRU}(\mathbf{X},\mathbf{h}_{0})=\mathrm{gru}(\mathbf{x}_{n},\dots,\mathrm{gru}(\mathbf{x}_{2},\mathrm{gru}(\mathbf{x}_{1},\mathbf{h}_{0})))$
(5)

where $\mathbf{X}$ stands for the matrix composed of input column vectors
$\mathbf{x}_{1},\ldots,\mathbf{x}_{n}$. Two or more GRU layers can be composed
into a stack:

$\mathrm{GRU}_{2}(\mathrm{GRU}_{1}(\mathbf{X},{\mathbf{h_{1}}}_{0}),{\mathbf{h_{2}}}_{0}).$
(6)

In our version of the Stacked GRU architecture we use residualized layers:

$\mathrm{GRU_{res}}(\mathbf{X},\mathbf{h}_{0})=\mathrm{GRU}(\mathbf{X},\mathbf{h}_{0})+\mathbf{X}$
(7)

Residual convolutional networks were introduced by ?), while ?) showed their
applicability to recurrent architectures. We adopt residualized layers here as
we observed they speed up learning in stacks of several GRU layers.

Our gated recurrent units use steep sigmoids for gate activations:

$\sigma_{s}(z)=\frac{1}{1+\exp(-3.75z)}$

and rectified linear units clipped between 0 and 5 for the unit activations:

$\sigma(z)=\mathrm{clip(0.5(z+\mathrm{abs}(z)),0,5)}$

There are two more components of our Phon GRU model: the phoneme encoding
layer, and mapping from the final state of the top GRU layer to the image
feature vector. The phoneme encoding layer is a simply a lookup table
$\mathbf{E}$ whose columns correspond to one-hot-encoded phoneme vectors. The
input phoneme $p_{t}$ of utterance $p$ at each step $t$ indexes into the
encoding matrix and produces the input column vector:

$\mathbf{x}_{t}=\mathbf{E}[:,p_{t}].$ (8)

Finally, we map the final state of the top GRU layer ${\mathbf{h_{K}}}_{n}$ to
the vector of image features using a fully connected layer:

$\hat{\mathbf{i}}=\mathbf{I}{\mathbf{h_{K}}}_{n}$ (9)

Our main interest lies in recurrent phoneme-level modeling. However, in order
to put the performance of the phoneme-level Phon GRU into perspective, we
compare it to two word-level models. Importantly, the word models should not
be seen as baselines, as they have access to word boundary and word identity
information not available to Phon GRU.

###  3.2 Word GRU

The architecture of this model is the same as Phon GRU with the difference
that we use words instead of phonemes as input symbols, use learnable word
embeddings instead of fixed one-hot phoneme encodings, and reduce the number
of layers in the GRU stack. See Section 4 for details.

###  3.3 Word Sum

The second model we use for comparison is a word-based non-sequential model,
consisting of a word embedding matrix, a vector sum operator, and a mapping to
the image feature vector:

$\hat{\mathbf{i}}=\mathbf{I}\sum_{t=1}^{n}\mathbf{E}[:,w_{t}]$ (10)

where $w_{t}$ is the word at position $t$ in the input utterance. This model
simply learns word embeddings which are then summed into a single vector and
projected to the target image vector. Thus this model does not have access to
word sequence information, and is a distributed analog of a bag-of-words
model.

##  4 Experiments

For all experiments, the models were trained on the training set of MS-COCO.
MS-COCO contains over 163,000 images accompanied by at least five captions by
human annotators. With an average of 7.7 labeled object instances per image,
images typically contain more objects than the captions mention, making
reference to the scene ambiguous. Textual input for the Phon GRU models was
transcribed automatically using the grapheme-to-phoneme functionality with the
default English voice of the eSpeak speech synthesis toolkit.111Available at
http://espeak.sourceforge.net Stress and pause markers were removed, as well
as word boundaries (after storing their position for use in experiments),
leaving only phoneme symbols. See Figure 2 for an example transcription.

Visual input for all models was obtained by forwarding images through the
16-layer convolutional neural network described in ?) pre-trained on Imagenet
[Russakovsky et al., 2014], and recording the activation vectors of the pre-
softmax layer. The z-score transformation was applied to these features to
ease optimization.

Most of the details of the three model types and training hyperparameters were
adopted from related work, and adapted via a limited amount of exploratory
experimentation. Exhaustive exploration of the search space was not feasible
due to the large number of adjustable settings in these models and their long
running time. Given the importance of depth for our purposes, we did
systematically explore the number of layers in the Phon GRU and Word GRU
models. A single layer is optimal for Word GRU. For Phon GRU, see Section 4.1
below. Other important settings were as follows:

  * •

All models: Implemented in Theano [Bastien et al., 2012], optimized with Adam
[Kingma and Ba, 2014], initial learning rate of 0.0002, minibatch size of 64,
gradient norm clipped to 5.0.

  * •

Word Sum: 1024-dimensional word embeddings, words with frequencies below 10
replaced by UNK token.

  * •

Word GRU: 1024-dimensional word embeddings, a single 1024 dimensional hidden
layer, words with frequencies below 10 replaced by UNK token.

  * •

Phon GRU: 1024-dimensional hidden layers.

###  4.1 Prediction of visual features

The models are trained and evaluated on the prediction of visual feature
vectors from captions. While our goal is not to develop an image retrieval
method, we use this task as it reflects the ability to extract visually
salient semantic information from language. For the experiments on the
prediction of visual features all models were trained on the training set of
MS-COCO. As validation and test data we used a random sample of 5000 images
each from the MS-COCO validation set.

Figure 4 shows the value of the validation average cosine distance between the
predicted visual vector and the target vector for three random initializations
of each of the model types.

The Phonetic GRU model is more sensitive to the initialization: one can
clearly distinguish three separate trajectories. The word-level models are
much less affected by random initialization. In terms of the overall
performance, the Phon GRU model falls between the Word Sum model and the Word
GRU model.

Figure 3: Value of the loss function on validation data during training. Three
random initialization of each model are shown.

Figure 4: Validation accuracy at 5 on the image retrieval task.

We also evaluated the models on how well they perform when used to search
images: for each validation sentence the model was used to predict the visual
vector. The image vectors in the validation data were then ranked by cosine
similarity to the predicted vector, and the proportion of times the correct
image was among the top 5 was reported. By correct image we mean the one which
the sentence was used to describe (even though often many other images are
also good matches to the sentence).

In Figure 4 we report the validation accuracies on this task for the two word-
level models, as well as for the Phon GRU model with different number of
hidden layers. We trained each model version with three random initializations
for each model setting, and evaluate after each epoch. We report the score of
the best epoch for each initialization. The overall ranking of the models
matches the direct evaluation of the loss function above: the phoneme-level
models are in between the two word-level models. Phon GRU with three hidden
layers is the best of the phoneme-level models.

In Table 2 we show the accuracies of the best version of each of the models
types on the test images; these are also the model versions used in all
subsequent experiments. The scores for the Word GRU are comparable to what ?)
report for their multitask Imaginet model, whose visual pathway has the same
structure, and who use the same data. More recently, ?) report substantially
higher scores for image search with a word-level GRU model, with the following
main differences from our setting: better image features, larger training set,
and a loss function optimized for the ranking task.222We have preliminary
results indicating that most of the analyses in the rest of Section 4 show the
same general pattern for phoneme models trained following the setting of ?).

###  4.2 Word boundary prediction

To explore the sensitivity of the Phon GRU model to linguistic structure at
the sub-word level, we investigated the encoding of information about word-
boundaries in the hidden layers. Logistic regression models were trained on
activation patterns of the hidden layers at all timesteps, with the objective
of identifying phonemes that preceded a word boundary. For comparison, we also
trained logistic regression models on n-gram data to perform the same tasks,
with positional phoneme n-grams in the range $1$-n. The location of the word
boundaries was taken from the eSpeak transcriptions, which mostly matches the
location of word boundaries according to conventional English spelling.
However, eSpeak models some coarticulation effects which sometimes leads to
word boundaries disappearing from the transcription. For example, bank of a
river is transcribed as [baNk @v@ rIv@].

All models were implemented using the LogisticRegression implementation from
Scikit-learn [Pedregosa et al., 2011] with L2-regularization. The random
samples of 5,000 images each that served as validation and test sets in the
visual feature prediction task were used as training and test sets. For the
models based on the activation patterns of the hidden layers, the z-score
transformation was applied to the activation values to ease optimization. The
optimal value of regularization parameter C was determined using GridSearchCV
with 5-fold cross validation on the training set, after which the model with
the optimal settings was trained on the full training sample.

Table 2 reports the scores on the test set. The proportion of phonemes
preceding a word boundary is 0.29, meaning that predicting no word boundary by
default would be correct in 0.71 of cases. At the highest hidden layer, enough
information about the word form is available for correct prediction in 0.82 of
cases – substantially above the majority baseline. The lower levels allow for
more accurate prediction of word boundaries: 0.86 at the middle hidden layer,
and 0.88 at the bottom level. Prediction scores of the logistic regression
model based on the activation patterns of the lowest hidden layer are
comparable to those of a bigram logistic regression model.

These results indicate that information on sub-word structure is only
partially encoded by Phon GRU, and is mostly absent by the time the signal
from the input propagates to the top layer. The bottom layer does learn to
encode a fair amount of word boundary information, but the prediction score
substantially below 100% indicates that it is rather selective.

Model | Acc @ 5 | Acc @ 10
---|---|---
Word Sum | 0.158 | 0.243
Word GRU | 0.205 | 0.306
Phon GRU | 0.180 | 0.276
Table 1: Image retrieval accuracy at 5 and at 10 on test data for the versions
of Word Sum, Word GRU and Phon GRU chosen by validation.

Model |  | Acc | Prec | Rec
---|---|---|---|---
Majority |  | 0.71 |  |
Phon GRU | Layer 1 | 0.88 | 0.82 | 0.78
| Layer 2 | 0.86 | 0.79 | 0.71
| Layer 3 | 0.82 | 0.74 | 0.60
n-gram |  n = 1 | 0.80 | 0.79 | 0.41
|  n = 2 | 0.87 | 0.79 | 0.78
|  n = 3 | 0.93 | 0.86 | 0.90
|  n = 4 | 0.95 | 0.90 | 0.93
Table 2: Prediction scores of logistic regression models based on activation
vectors of Phon GRU and on positional n-grams

###  4.3 Word similarity

To understand the encoding of semantic information in Phon GRU, we analyzed
the cosine similarity of activation vectors for word pairs from the MEN Test
Collection [Bruni et al., 2014]. The MEN dataset contains 3,000 pairs of
English words with semantic similarity judgements on a 50-point scale, which
were obtained through crowd-sourcing.333The MEN dataset is available at
http://clic.cimec.unitn.it/~elia.bruni/MEN For each word pair in the MEN
dataset, the words were transcribed phonetically using eSpeak and then fed to
Phon GRU individually. For comparison, the words were also fed to Word GRU and
Word Sum. Word pair similarity was quantified as the cosine similarity between
the activation patterns of the hidden layers at the end-of-sentence symbol. In
contrast to Word GRU and Word Sum, Phon GRU has access to the sub-word
structure. To explore the role of phonemic form in word similarity, a measure
of phonemic difference was included: the Levenshtein distance between the
phonetic transcriptions of the two words, normalized by the length of the
longer transcription.

Table 4 shows Spearman's rank correlation coefficient between human similarity
ratings from the MEN dataset and cosine similarity at the last timestep for
all hidden layers. In all layers, the cosine similarities between the
activation vectors for two words are significantly correlated with human
similarity judgements. The strength of the correlation differs considerably
between the layers, ranging from 0.09 in the first layer to 0.28 in the
highest hidden layer. The second column in Table 4 shows the correlations when
only taking into account the 1283 word pairs of which both words appear at
least 100 times in the training set of MS-COCO. Correlations for both Word GRU
and Word SUM are considerably higher than for Phon GRU. This is expected given
that these are word level models with explicit word-embeddings, while Phon GRU
builds word representations by forwarding phoneme-level input through several
layers of processing.

| All words | Frequent words
---|---|---
Phon GRU Layer 1 | 0.09 | 0.12
Layer 2 | 0.21 | 0.33
Layer 3 | 0.28 | 0.45
Word GRU | 0.48 | 0.60
Word Sum | 0.42 | 0.56
Table 3: Spearman's correlation coefficient between word-word cosine
similarity and human similarity judgements. All correlations significant at p
$<1\mathrm{e}{-4}$. Frequent words appear at least 100 times in the training
data.

Layer | $\rho$
---|---
1 | $-0.30$
2 | $-0.24$
3 | $-0.15$
Table 4: Spearman's rank correlation coefficient between Phon GRU cosine
similarity and phoneme-level edit distance. All correlations significant at p
$<1\mathrm{e}{-15}$.

Table 4 shows Spearman's rank correlation coefficient between the edit
distance and the cosine similarity of activation vectors at the hidden layers
of Phon GRU. As expected, edit distance and cosine similarity of the
activation vectors are negatively correlated: words which are more similar in
form are also more similar according to the model.444Note that in the MEN
dataset, meaning and word form are also (weakly) correlated: human similarity
judgements and edit distance are correlated at $-0.08$ ($p<1\mathrm{e}{-5}$).

The negative correlation between edit distances and cosine similarities is
strongest at the lowest hidden layer and weakest, though still present and
stronger than for human judgements, at the third hidden layer.

The correlations of cosine similarities with edit distance on the one hand,
and human similarity rating on the other hand, indicate that the different
hidden layers reflect increasing levels of representation: whereas at the
lowest level mostly encodes information about form, the highest layer mostly
encodes semantic information.

###  4.4 Position of shared substrings

Here we quantify the time-scale at which information is retained in the
different layers of Phon GRU. We looked at the location of phoneme strings
shared by sentences and their nearest neighbors in the 5,000-image validation
sample. We determined each sentence's nearest neighbor for each hidden layer
in Phon GRU. The nearest neighbour is the sentence for which the activation
vector at the end of sentence symbol has the smallest cosine distance to the
activation vector of the original sentence. The position of matching
substrings is the average position in the original sentence of symbols in
substrings that are shared by the neighbor sentences, counted from the end of
the sentence. A high mean average substring position thus means that the
shared substring(s) appear early in the sentence. This gives an indirect
measure of the timescale at which the different layers operate. Table 6 shows
an example.

As can be seen in Table 6, the average position of shared substrings in
neighbor sentences is closest to the end for the first hidden layer and moves
towards the beginning of the sentence for the second and third hidden layer.
This indicates a difference between the layers with regards to the timescale
they represent. Whereas in the lowest layer only information from the latest
timesteps is present, the higher layers retain the input signal over longer
timescales.

Layer 1
---
A metallic bench on a path in the park
A man riding a bicycle on a path in a park
Layer 3
A metallic bench on a path in the park
A stone park bench sitting in an empty green park
Table 5: An illustrative sentence with its nearest neighbour at layer 1 and
layer 3. For readability, sentences are displayed in conventional spelling,
and only highlight matching substrings of length $\geq 3$. In reality we used
phonetic transcriptions to compute shared substring positions, and substrings
of all lengths.

Layer | Mean position
---|---
1 | 12.1
2 | 14.9
3 | 16.8
Table 6: Average position of phonemes in shared substrings between nearest
neighbour sentences according to Phon GRU representations at the different
layers. Positions are indexed from end of string.

##  5 Future work

Although our analyses show a clear pattern of short-timescale information in
the lower layers and larger dependencies in the higher layers, the third layer
still encodes information about the phonetic form: its activation patterns
were predictive of word boundaries, and similarities between word pairs at
this level were more strongly correlated with edit distance than human
similarity judgements are. It would be interesting to investigate exactly what
information that is, and to what extent it is analogous to language
representation in the mind of human speakers. In humans both word phonological
form and word meaning can act as primes, which is somewhat reminiscent of the
behavior of our model.

Finally, we would like to take the next step towards grounded learning of
language from raw perceptual input, and apply models similar to the one
described here to acoustic speech signal coupled with visual input. We expect
this to be a challenging but essential endeavor.

## References

  * [Bastien et al., 2012]  Frédéric Bastien, Pascal Lamblin, Razvan Pascanu, James Bergstra, Ian J. Goodfellow, Arnaud Bergeron, Nicolas Bouchard, and Yoshua Bengio.  2012\.  Theano: new features and speed improvements.  Deep Learning and Unsupervised Feature Learning NIPS 2012 Workshop.
  * [Bernardi et al., 2016]  Raffaella Bernardi, Ruket Cakici, Desmond Elliott, Aykut Erdem, Erkut Erdem, Nazli Ikizler-Cinbis, Frank Keller, Adrian Muscat, and Barbara Plank.  2016\.  Automatic description generation from images: A survey of models, datasets, and evaluation measures.  arXiv preprint arXiv:1601.03896.
  * [Bruni et al., 2014]  Elia Bruni, Nam-Khanh Tran, and Marco Baroni.  2014\.  Multimodal distributional semantics.  Journal of Artificial Intelligence Research (JAIR), 49:1–47.
  * [Cho et al., 2014]  Kyunghyun Cho, Bart van Merriënboer, Dzmitry Bahdanau, and Yoshua Bengio.  2014\.  On the properties of neural machine translation: Encoder-decoder approaches.  In Eighth Workshop on Syntax, Semantics and Structure in Statistical Translation (SSST-8).
  * [Chrupała et al., 2015]  Grzegorz Chrupała, Akos Kádár, and Afra Alishahi.  2015\.  Learning language through pictures.  In Proceedings of the 53rd Annual Meeting of the Association for Computational Linguistics.
  * [Chrupała, 2013]  Grzegorz Chrupała.  2013\.  Text segmentation with character-level text embeddings.  In Proceedings of the ICML workshop on Deep Learning for Audio, Speech and Language Processing.
  * [Chung et al., 2014]  Junyoung Chung, Caglar Gulcehre, KyungHyun Cho, and Yoshua Bengio.  2014\.  Empirical evaluation of gated recurrent neural networks on sequence modeling.  In NIPS 2014 Deep Learning and Representation Learning Workshop.
  * [Chung et al., 2016]  Junyoung Chung, Kyunghyun Cho, and Yoshua Bengio.  2016\.  A character-level decoder without explicit segmentation for neural machine translation.  arXiv preprint arXiv:1603.06147.
  * [Cunillera et al., 2010]  Toni Cunillera, Matti Laine, Estela Càmara, and Antoni Rodríguez-Fornells.  2010\.  Bridging the gap between speech segmentation and word-to-world mappings: Evidence from an audiovisual statistical learning task.  Journal of Memory and Language, 63(3):295 – 305.
  * [Elman, 1990]  Jeffrey L Elman.  1990\.  Finding structure in time.  Cognitive science, 14(2):179–211.
  * [Fazly et al., 2010]  Afsaneh Fazly, Afra Alishahi, and Suzanne Stevenson.  2010\.  A probabilistic computational model of cross-situational word learning.  Cognitive Science: A Multidisciplinary Journal, 34(6):1017–1063.
  * [Frank et al., 2007]  Michael C. Frank, Noah D. Goodman, and Joshua B. Tenenbaum.  2007\.  A Bayesian framework for cross-situational word-learning.  In Advances in Neural Information Processing Systems, volume 20.
  * [Glicksohn and Cohen, 2013]  Arit Glicksohn and Asher Cohen.  2013\.  The role of cross-modal associations in statistical learning.  Psychonomic Bulletin & Review, 20(6):1161–1169.
  * [Harwath and Glass, 2015]  David Harwath and James Glass.  2015\.  Deep multimodal semantic embeddings for speech and images.  arXiv:1511.03690.
  * [He et al., 2015]  Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.  2015\.  Deep residual learning for image recognition.  arXiv:1512.03385.
  * [Hermans and Schrauwen, 2013]  Michiel Hermans and Benjamin Schrauwen.  2013\.  Training and analysing deep recurrent neural networks.  In Advances in Neural Information Processing Systems, pages 190–198.
  * [Iwahashi, 2003]  Naoto Iwahashi.  2003\.  Language acquisition through a human–robot interface by combining speech, visual, and behavioral information.  Information Sciences, 156(1):109–121.
  * [Kádár et al., 2016]  Ákos Kádár, Grzegorz Chrupała, and Afra Alishahi.  2016\.  Representation of linguistic form and function in recurrent neural networks.  CoRR, abs/1602.08952.
  * [Karpathy et al., 2015]  Andrej Karpathy, Justin Johnson, and Fei-Fei Li.  2015\.  Visualizing and understanding recurrent networks.  arXiv preprint arXiv:1506.02078.
  * [Kingma and Ba, 2014]  Diederik P. Kingma and Jimmy Ba.  2014\.  Adam: A method for stochastic optimization.  CoRR, abs/1412.6980.
  * [Kiros et al., 2015]  Ryan Kiros, Yukun Zhu, Ruslan R Salakhutdinov, Richard Zemel, Raquel Urtasun, Antonio Torralba, and Sanja Fidler.  2015\.  Skip-thought vectors.  In Advances in Neural Information Processing Systems, pages 3276–3284.
  * [Landauer et al., 1998]  Thomas K Landauer, Peter W Foltz, and Darrell Laham.  1998\.  An introduction to latent semantic analysis.  Discourse processes, 25(2-3):259–284.
  * [Lazaridou et al., 2015]  Angeliki Lazaridou, Nghia The Pham, and Marco Baroni.  2015\.  Combining language and vision with a multimodal skip-gram model.  In Proceedings of NAACL HLT 2015 (2015 Conference of the North American Chapter of the Association for Computational Linguistics - Human Language Technologies).
  * [Lazaridou et al., 2016]  Angeliki Lazaridou, Grzegorz Chrupała, Raquel Fernández, and Marco Baroni.  2016\.  Multimodal semantic learning from child-directed input.  In The 15th Annual Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies.
  * [Lin et al., 2014]  Tsung-Yi Lin, Michael Maire, Serge Belongie, James Hays, Pietro Perona, Deva Ramanan, Piotr Dollár, and C Lawrence Zitnick.  2014\.  Microsoft coco: Common objects in context.  In Computer Vision–ECCV 2014, pages 740–755. Springer.
  * [Ling et al., 2015]  Wang Ling, Tiago Luís, Luís Marujo, Ramón Fernandez Astudillo, Silvio Amir, Chris Dyer, Alan W Black, and Isabel Trancoso.  2015\.  Finding function in form: Compositional character models for open vocabulary word representation.  In Proceedings of EMNLP.
  * [Oord et al., 2016]  Aaron van den Oord, Nal Kalchbrenner, and Koray Kavukcuoglu.  2016\.  Pixel recurrent neural networks.  arXiv preprint arXiv:1601.06759.
  * [Pedregosa et al., 2011]  Fabian Pedregosa, Gaël Varoquaux, Alexandre Gramfort, Vincent Michel, Bertrand Thirion, Olivier Grisel, Mathieu Blondel, Peter Prettenhofer, Ron Weiss, Vincent Dubourg, Jake Vanderplas, Alexandre Passos, David Cournapeau, Matthieu Brucher, Matthieu Perrot, and Édouard Duchesnay.  2011\.  Scikit-learn: Machine learning in Python.  Journal of Machine Learning Research, 12:2825–2830.
  * [Plank et al., 2016]  Barbara Plank, Anders Søgaard, and Yoav Goldberg.  2016\.  Multilingual part-of-speech tagging with bidirectional long short-term memory models and auxiliary loss.  In Proceedings of ACL 2016. Association for Computational Linguistics (ACL).
  * [Räsänen and Rasilo, 2015]  Okko Räsänen and Heikki Rasilo.  2015\.  A joint model of word segmentation and meaning acquisition through cross-situational learning.  Psychological review, 122(4):792.
  * [Roy and Pentland, 2002]  Deb K Roy and Alex P Pentland.  2002\.  Learning words from sights and sounds: a computational model.  Cognitive Science, 26(1):113 – 146.
  * [Russakovsky et al., 2014]  Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael Bernstein, Alexander C. Berg, and Li Fei-Fei.  2014\.  ImageNet Large Scale Visual Recognition Challenge.
  * [Simonyan and Zisserman, 2014]  Karen Simonyan and Andrew Zisserman.  2014\.  Very deep convolutional networks for large-scale image recognition.  CoRR, abs/1409.1556.
  * [Siskind, 1996]  Jeffrey M. Siskind.  1996\.  A computational study of cross-situational techniques for learning word-to-meaning mappings.  Cognition, 61(1-2):39–91.
  * [Synnaeve et al., 2014]  Gabriel Synnaeve, Maarten Versteegh, and Emmanuel Dupoux.  2014\.  Learning words from images and speech.  In NIPS Workshop on Learning Semantics, Montreal, Canada.
  * [Thiessen, 2010]  Erik D Thiessen.  2010\.  Effects of visual information on adults’ and infants’ auditory statistical learning.  Cognitive Science, 34(6):1093–1106.
  * [Vendrov et al., 2016]  Ivan Vendrov, Ryan Kiros, Sanja Fidler, and Raquel Urtasun.  2016\.  Order-embeddings of images and language.  In ICLR.
  * [Xie et al., 2016]  Ziang Xie, Anand Avati, Naveen Arivazhagan, Dan Jurafsky, and Andrew Y Ng.  2016\.  Neural language correction with character-based attention.  arXiv preprint arXiv:1603.09727.
  * [Yurovsky et al., 2012]  Daniel Yurovsky, Chen Yu, and Linda B Smith.  2012\.  Statistical speech segmentation and word learning in parallel: scaffolding from child-directed speech.  Frontiers in Psychology, 3(374).


# On pseudo-absorbing primary multiplication modules
over pullback rings††thanks: Key Words: Pseudo-absorbing primary
multiplication modules, 2-absorbing primary ideals, Separated modules,
Pullback ring. ††thanks:  2010 Mathematics Subject Classification: 13C05,
16D70, 03C45.

M.J. Nikmehra , R. Nikandishb and A. Yassinea

aFaculty of Mathematics, K.N. Toosi University of Technology,
P.O. BOX 16315-1618, Tehran, Iran
<EMAIL_ADDRESS><EMAIL_ADDRESS>
bDepartment of Mathematics, Jundi-Shapur University of Technology,
P.O. BOX 64615-334, Dezful, Iran
<EMAIL_ADDRESS>
Corresponding author

###### Abstract

A famous result due to L. S. Levy provides a classification of all finitely
generated indecomposable modules over Dedekind-like rings. This motivates us
to outline an approach to the classification of indecomposable pseudo-
absorbing primary multiplication modules with finite-dimensional top over
certain kinds of pullback rings. In this paper, we give a complete
classification, up to isomorphism, of all indecomposable pseudo-absorbing
primary multiplication modules with finite-dimensional top over a pullback of
two valuation domains with the same residue field. We also find a connection
between pseudo-absorbing primary multiplication modules and pure-injective
modules over such domains.

##  1 Introduction

A basic problem in mathematics is to classify arbitrary modules one is
studying up to isomorphism. A lesson many authors learned from representation
types (finite, tame, or wild) is that while this is almost always impossible
(see [1, 29, 30] and [31, Chapter 19]), it is sometimes possible, and very
useful, to classify certain full subcategories of the category one is working
in. In particular, the pure-injective modules and also indecomposable
representable modules have been classified [8, 9]. Indeed, the classification
of the complete theories of modules reduces to classifying the pure-
injectives. Notably, there are important classifications of modules like the
classification of finite-dimensional and finitely generated modules, in many
cases this classification can be extended to classify pure-injective modules.
In fact, there is a connection between infinitely generated pure-injective
modules and finitely generated modules in many cases (see [5, 27, 28]). The
purpose of this paper is to introduce a class of modules called pseudo-
absorbing primary multiplication modules which form an interesting subclass of
pure-injective modules.

Modules over pullback rings have been studied by several authors (see [3, 13,
8, 10, 12, 18, 19] and [32]). The first important contribution in this
direction is due to Levy [21], who classified all finitely generated
indecomposable modules over Dedekind-like rings. Later, this classification
was extended by Klingler [19] to lattices over certain non-commutative
Dedekind-like rings, and then Haefner and Klingler classified lattices over
certain non-commutative pullback rings, which they called special quasi-triads
[15, 16]. The common of all these classifications is the reduction to a
“matrix problem” over a division ring [6].

We assume throughout this paper that all rings are commutative with identity.
Recall the construction: if $R_{1}$ and $R_{2}$ are two discrete valuation
domains with maximal ideals $P_{1}=Rp_{1}$ and $P_{2}=Rp_{2}$ respectively,
then
$R=(R_{1}\xrightarrow{\mathcal{V}_{1}(x)}\overline{R}\xleftarrow{\mathcal{V}_{2}(x)}R_{2})$
is the pullback ring of $R_{1}$ and $R_{2}$ with a unique maximal ideal
$P=P_{1}\oplus P_{2}$. Note that $\overline{R}\cong R/P\cong R_{1}/P_{1}\cong
R_{2}/P_{2}$ is a field. This means that $R$ is a commutative Noetherian local
ring with unique maximal ideal $P$. So it is easy to see that $P_{1}$ and
$P_{2}$ are the other prime ideals of $R$. The elements of this ring are of
the form $a=(r,s)$ where $r\in R_{1}$ and $s\in R_{2}$ such that
$\mathcal{V}_{1}(r)=\mathcal{V}_{2}(s)$. If $r$ and $s$ are nonzero, then
$a=(p_{1}^{n},p_{2}^{m})$ for some positive integers $m,n$, which means that
ann$(a)=0$. Therefore $Ra\cong R$. But if $a=(0,p_{2}^{m})$
($a=(p_{1}^{n},0)$) for some positive integers $m,n$, then ann$(a)=P_{1}\oplus
0$ (ann$(a)=0\oplus P_{2}$), and thus $R/(P_{1}\oplus 0)\cong
R(0,p_{2}^{m})\cong R_{2}$ ($R/(0\oplus P_{2})\cong R(p_{1}^{n},0)\cong
R_{1}$). The other ideals $I$ of $R$ are of the form $I=P_{1}^{n}\oplus
P_{2}^{m}=(P^{n}_{1},P^{m}_{2})=(<p^{n}_{1}>,<p^{m}_{2}>)$ for some positive
integers $m,n$.

Following some ideas and a technique introduced in S. Ebrahimi Atani et al.
Pseudo-absorbing multiplication modules over a pullback ring (J. Pure Appl.
Algebra 222 (2018), 3124–3136), in this paper we introduce a class of modules
called pseudo-absorbing primary multiplication modules which form a subclass
of pure-injective modules (see Definition 2.1), and contains the class of
pseudo-absorbing multiplication modules [13] and the class of pseudo-prime
multiplication modules [12]. We are mainly interested to study these modules
in the two following cases: where $R$ is a Dedekind domain or a pullback ring
of two local Dedekind domains. In section 2, a complete description of pseudo-
absorbing primary multiplication modules over a discrete valuation domain and
some of their basic properties are given. The classification of these modules
is divided into two steps: indecomposable separated pseudo-absorbing primary
multiplication and indecomposable non-separated pseudo-absorbing primary
multiplication $R$-modules. In section 3, the indecomposable separated pseudo-
absorbing primary multiplication modules over a pullback of two local Dedekind
domains with finite-dimensional top over $R/Rad(R)$ are completely described.
In section 4, we use the list of separated pseudo-absorbing primary
multiplication modules obtained in section 3 and results of Levy [21, 22] on
the possibilities of amalgamating finitely generated separated modules to show
that non-separated indecomposable pseudo-absorbing primary multiplication
$R$-modules with finite-dimensional top are precisely finite direct sums of
separated indecomposable separated pseudo-absorbing primary multiplication
$R$-modules. We will see also that non-separated modules may be represented by
certain amalgamation chains of indecomposable separated pseudo-absorbing
primary multiplication modules and this abutment corresponds to amalgamation
in the socles of these separated pseudo-absorbing primary multiplication
modules.

For completeness, we define the concepts that will be needed. A proper ideal
$I$ of a commutative ring $R$ is called $2$-absorbing primary if whenever
$a,b,c\in R$ and $abc\in I$, then $ab\in I$ or $ac\in\sqrt{I}$ or
$bc\in\sqrt{I}$ [2]. This notion was extended to the concept of $2$-absorbing
primary submodule in [25] which is a generalization of prime submodule with
the following definition, if whenever $a,b\in R$ and $m\in M$ and $abm\in N$
for some proper submodule $N$ of $M$, then $am\in N$ or $bm\in N$ or
$ab\in\sqrt{(N:M)}$. An $R$-module $M$ is called multiplication if for each
submodule $N$ of $M$, $N=IM$, for some ideal $I$ of $R$ [14]. A submodule $N$
of an $R$-module $M$ is said to be pure if any finite system of equations over
$N$ which is solvable in $M$ is also solvable in $N$. A submodule $N$ of an
$R$-module $M$ is said to be relatively divisible in $M$ if $rN=N\cap rM$ for
all $r\in R$ [26]. A module $M$ is said to be pure-injective if it has the
injective property relative to all pure exact sequences [8, 26]. An $R$-module
$S$ is said to be separated if there exist $R_{i}$-modules $S_{i}$, $i=1,2$,
such that $S$ is a submodule of $S_{1}\oplus S_{2}$. Let $R$ be a pullback
ring as above. Equivalently, $S$ is separated if and only if $P_{1}S\cap
P_{2}S=0$ [20]. Let $M$ be an $R$-module. A separated representation of $M$ is
an epimorphism $\varphi:S\rightarrow M$ of $R$-modules where $S$ is separated
and $\varphi$ is an $R$-homomorphism, and if $\varphi$ admits a factorization
$\varphi=(S\xrightarrow{f}S^{\prime}\rightarrow M)$ such that $S^{\prime}$ is
separated, then $f$ is one-to-one. The module $K=$ Ker$(\varphi)$ is an
$\overline{R}$-module, since $\overline{R}=R/P$ and $PK=0$ [20, Proposition
2.3]. An exact sequence $0\rightarrow
K\xrightarrow{i}S\xrightarrow{\varphi}M\rightarrow 0$ of $R$-modules with $S$
separated and $K$ an $\overline{R}$-module is a separated representation of
$M$ if and only if $P_{i}S\cap K=0$ for each $i$ and $K\subseteq PS$ [22,
Proposition 2.3]. It follows from [20, Theorem 2.8] that every module $M$ has
a separated representation, which is unique up to isomorphism. For undefined
notation and terminology, the reader is referred to [8, 9, 13].

##  2 Some properties of pseudo-absorbing primary multiplication modules

In this section, the notion of pseudo-absorbing primary multiplication modules
over a discrete valuation domain is introduced and some of their basic
properties are given.

###### Definition 2.1

Let $R$ be a ring and $M$ be an $R$-module. A proper submodule $N$ of $M$ is
said to be pseudo-absorbing primary if $(N:_{R}M)$ is a $2$-absorbing primary
ideal of $R$. Furthermore, $M$ is said to be a pseudo-absorbing primary
multiplication module if for every pseudo-absorbing primary submodule $N$ of
$M$, there is an ideal $I$ of $R$ such that $N=IM$.

It is easy to see that if $N$ is a pseudo-absorbing primary submodule of a
pseudo-absorbing primary multiplication module $M$, then there exists an ideal
$I$ of $R$ such that $N=IM$, and so $N=IM\subseteq(N:_{R}M)M\subseteq N$.
Hence $N=(N:_{R}M)M$ for every pseudo-absorbing primary submodule $N$ of $M$.
Since every prime ideal is a $2$-absorbing ideal and every $2$-absorbing ideal
is a $2$-absorbing primary ideal, one can easily see that the class of pseudo-
absorbing primary multiplication modules contains the class of pseudo-
absorbing multiplication modules and the class of pseudo-absorbing
multiplication modules contains the class of pseudo-prime multiplication
modules.

The following example shows that a pseudo-absorbing primary submodule of $M$
need not be a primary (prime) or pseudo-absorbing or $2$-absorbing primary
submodule of $M$.

###### Example 2.2

$(1)$ Let $M=\mathbb{Z}_{16}=\\{0,1,2,\ldots,15\\}$ be an $R$-module such that
$R=\mathbb{Z}$ and $N=\\{0,8\\}$ a submodule of $M$. Then
$(N:_{R}M)=8\mathbb{Z}$ and $\sqrt{(N:_{R}M)}=2\mathbb{Z}$. It is easy to see
that $N$ is a $2$-absorbing primary submodule of $M$, and so it is a pseudo-
absorbing primary submodule of $M$. Now, $2\cdot 2\cdot 2\in(N:_{R}M)$ but
$2\cdot 2\notin(N:_{R}M)$. Hence, $N$ is a pseudo-absorbing primary submodule
of $M$ which is not pseudo-absorbing.

$(2)$ Let $N=6\mathbb{Z}$ be a submodule of a $\mathbb{Z}$-module
$\mathbb{Z}$. Since $N$ is a $2$-absorbing primary submodule of $M$, $N$ is a
pseudo-absorbing primary submodule of $M$. But it is not a primary (prime)
submodule, as $2\cdot 3\in N$, whereas $2\notin N$ and
$3\notin\sqrt{(N:_{R}M)}$.

$(3)$ It is easy to see that if $N$ is a $2$-absorbing primary submodule of an
$R$-module $M$, then $N$ is a pseudo-absorbing primary submodule of $M$.
Asuume that $R$ is a discrete valuation domain with maximal deal $P=Rp$. It
follows from [9, Lemma 2.6] that every non-zero proper submodule $L$ of
$E=E(R/P)$, the injective hull of $R/P$, is of the form
$L=A_{n}=(0:_{E}P^{n})$ $(n\geq 1)$, $L=A_{n}=Ra_{n}$ and $PA_{n+1}=A_{n}$.
One can see that no $A_{n}$ is a 2-absorbing primary submodule of $E$, for
otherwise, if $n$ is a positive integer, then $P^{2}A_{n+2}=A_{n}$, and
neither $PA_{n+2}=A_{n+1}\subseteq A_{n}$ nor
$P^{2}\subseteq\sqrt{(A_{n}:E)}=0$. Hence no $A_{n}$ is a 2-absorbing primary
submodule of $E$. Now, $(A_{n}:E)=0$ implies $A_{n}$ is a pseudo-absorbing
primary submodule for all $n\geq 1$, and thus a pseudo-absorbing primary
submodule of $M$ need not be a $2$-absorbing primary submodule of $M$.

###### Proposition 2.3

Suppose that $R$ is a ring and $N\subseteq K$ are submodules of an $R$-module
$M$. Then the following statements hold:

$(1)$ If $K$ is a pseudo-absorbing primary submodule of $M$, then $K/N$ is
apseudo-absorbing primary submodule of $M/N$.

$(2)$ If $I$ is an ideal of $R$ such that $I\subseteq(0:_{R}M)$ and $K$ is a
pseudo-absorbing primary submodule of $M$, then $K$ is a pseudo-absorbing
primary submodule of $M$ as an $R/I$-module.

$(3)$ If $K$ is a $2$-absorbing primary and pseudo-prime submodule of $M$,
then $K$ is a $2$-absorbing submodule of $M$. The converse is true whenever
prime ideals of $R$ are comparable; in particular, $R$ is local with maximal
ideal $M$ and for every minimal prime ideal $P$ over a $2$-absorbing ideal
$I$, we have $IM=P$.

###### Proof.

$(1)$ This is obvious because $(K:_{R}M)=(K/N:_{R}M/N)$.

$(2)$ Let $(K:_{R}M)=Q$ for some $2$-absorbing primary ideal $Q$ of $R$. Since
$IM=0$, $I\subseteq Q$, and hence an inspection will show that
$(K:_{R/I}M)=Q/I$. Thus $K$ is a pseudo-absorbing primary submodule of $M$ as
an $R/I$-module.

$(3)$ Suppose that $K$ is a $2$-absorbing submodule of $M$. It follows from
[4, Theorem 2.3] that $K$ is a pseudo-prime submodule of $M$. Conversely,
suppose that $K$ is a $2$-absorbing primary and pseudo-prime submodule of $M$,
and $abm\in K$ for some $a,b\in R$ and $m\in M$. Suppose also that
$am,bm\notin K$. Then $(ab)^{n}\in(K:_{R}M)$ for some positive integer $n$,
since $K$ is a $2$-absorbing primary submodule of $M$. But $K$ is a pseudo-
prime submodule of $M$, hence $ab\in(K:_{R}M)$. Thus $K$ is a $2$-absorbing
submodule of $M$.  $\Box$

###### Proposition 2.4

Suppose that $f:R\to R^{\prime}$ is a homomorphism and $M$ is a pseudo-
absorbing primary multiplication $R$-module and suppose also that $N$ is a
submodule of $M$. Then the following statements hold:

$(1)$ For every ideal $I$ of $R$ such that $I\subseteq(N:_{R}M)$ and $N$ is a
non-zero of $M$, $M/N$ is a pseudo-absorbing primary multiplication
$R/I$-module. In particular, $M/N$ is a pseudo-absorbing primary
multiplication $R$-module.

$(2)$ Every direct summand of $M$ is a pseudo-absorbing primary multiplication
submodule of $M$.

$(3)$ If $I\subseteq(0:_{R}M)$ where $I$ is an ideal of $R$, then $M$ is a
pseudo-absorbing primary multiplication $R$-module if and only if $M$ is
pseudo-absorbing primary multiplication as an $R/I$-module.

$(4)$ If $M^{\prime}$ is a pseudo-absorbing primary multiplication
$R^{\prime}$-module, then $M^{\prime}$ is a pseudo-absorbing primary
multiplication $R$-module.

###### Proof.

$(1)$ Suppose that $K/N$ is a pseudo-absorbing primary submodule of $M/N$. It
follows from Proposition 2.3 $(1)$, $K$ is a pseudo-absorbing primary
submodule of $M$, and hence $K=(K:M)M$. This means that
$I\subseteq(N:_{R}M)\subseteq(K:_{R}M)=J$. An inspection shows that
$K/N=(J/I)(M/N)$. For the“in particular” statement take $I=0$.

$(2)$ This follows from part $(1)$.

$(3)$ It follows from Proposition 2.3 $(2)$ that $N$ is a pseudo-absorbing
primary submodule of $M$ if and only if $N$ is a pseudo-absorbing primary
$R/I$-submodule of $M$. Now the assertion follows from the fact that
$(N:_{R}M)=(N:_{R/I}M)$.

$(4)$ Suppose that $f:R\to R^{\prime}$ is a surjective homomorphism and
$M^{\prime}$ is a pseudo-absorbing primary multiplication $R^{\prime}$-module.
Suppose also that $N$ is a pseudo-absorbing primary $R$-submodule of
$M^{\prime}$. Then $N=IM^{\prime}$ for some ideal $I$ of $R^{\prime}$. Since
$f$ is a surjective homomorphism, there is and ideal $J$ of $R$ such that
$J=f^{-1}(I)$ with $f(J)=I$. Then $JM^{\prime}=f(J)M^{\prime}=IM^{\prime}=N$.
Hence $M^{\prime}$ is a pseudo-absorbing primary multiplication $R$-module.
$\Box$

In the following result, we show that if $R$ is a discrete valuation domain
with maximal deal $P$, then the injective hull of $R/P$ and the field of
fractions of $R$ are not pseudo-absorbing primary submodules.

###### Proposition 2.5

Suppose that $R$ is a discrete valuation domain with maximal deal $P$. Then
the injective hull $E(R/P)$ of $R/P$ and the field of fraction $Q(R)$ of $R$
are not pseudo-absorbing primary multiplication modules. Furthermore, the
cyclic modules $R$ and $R/P^{n}$ are pseudo-absorbing primary multiplication
modules.

###### Proof.

First we show that every divisible pseudo-absorbing primary multiplication
module $M$ over an integral domain $R$ is a simple module. Suppose that $N$ is
a proper submodule of $M$. Since $M$ is divisible, $(N:_{R}M)=0$, and so
$N=(N:_{R}M)M=0_{M}=0$. Thus $M$ is a simple module. Clearly $E(R/P)$ and
$Q(R)$ are divisible modules and by [9, Lemma 2.6], $E(R/P)$ has nonzero
proper submodules and the submodule $N=\\{r/1:r\in R\\}$ is a nonzero proper
submodule of $Q(R)$. Therefore $E(R/P)$ and $Q(R)$ are not simple, and hence
the injective hull $E(R/P)$ of $R/P$ and the field of fraction $Q(R)$ of $R$
are not pseudo-absorbing primary multiplication modules. Now, since the cyclic
modules $R$ and $R/P^{n}$ are multiplication, they are also pseudo-absorbing
primary multiplication modules.  $\Box$

The following result contains a complete list of indecomposable pseudo-
absorbing primary multiplication modules over local Dedekind domains.

###### Theorem 2.6

Suppose that $R$ is a discrete valuation domain with maximal ideal $P=Rp$.
Then the indecomposable pseudo-absorbing primary multiplication modules over
$R$, up to isomorphism, are:

$(1)$ $R$;

$(2)$ $R/P^{n}$, $n\geq 1$ the indecomposable torsion modules.

###### Proof.

It follows from [8, Proposition 1.3] and Proposition 2.5 that the modules $R$
and $R/P^{n}$ ($n\geq 1$) are indecomposable pseudo-absorbing primary
multiplication. It remains to show that there are no more indecomposable
pseudo-absorbing primary multiplication $R$-modules. For otherwise we assume
that $M$ is an indecomposable pseudo-absorbing primary multiplication module,
and suppose that $0\neq a\in M$. Suppose also that $h(a)=$ sup$\\{n:a\in
P^{n}M\\}$ (this means that $h(a)$ is either a nonnegative integer or
$\infty$). One can see that either $(0:_{R}a)=P^{m}$ or $(0:_{R}a)=0$, since
if $(0:_{R}a)=P^{m+1}$, $p^{m}a\neq 0$ and $pp^{m}a=0$, and so we can take $a$
such that $(0:_{R}a)=P$ or $0$. We consider three cases.
Case one: If $h(a)=n$ and $(0:_{R}a)=P$, then $a=p^{n}b$ for some positive
integer $n$ and $b\in M$. Therefore $Rb\cong R/P^{n+1}$ is a pseudo-absorbing
primary multiplication $R$-module. Hence by a similar argument like that in
[7, Theorem 2.12 Case 1], the submodule $Rb$ of $M$ is pure. But $Rb$ is a
submodule of bounded order of $M$, hence [17, Theorem 5] implies that $Rb$ is
a direct summand of $M$. Thus $M=Rb\cong R/P^{n+1}$.
Case two: If $h(a)=n$ and $(0:_{R}a)=0$, then $a=p^{n}b$ for some positive
integer $n$ and $b\in M$. Therefore $(0:_{R}b)=0$, and hence $Rb\cong R$. By a
similar argument like that in Case 1, the submodule $Rb$ of $M$ is pure. Hence
by a similar argument like that in [7, Theorem 2.12 Case 2], there is a
positive integer $t$ such that $R\cong Rb=P^{t}M=M$.
Case three: Assume that $h(a)=\infty$. If $(0:_{R}a)=P$, then as in [7,
Theorem 2.12 Case 4] we have $M\cong E(R/P)$, a contradiction, by Proposition
2.5. If $(0:_{R}a)=0$, then as in [7, Theorem 2.12 Case 3] we have $M\cong
Q(R)$, a contradiction, by Proposition 2.5.  $\Box$

We close this section with the following result.

###### Corollary 2.7

Suppose that $R$ is a discrete valuation domain with maximal ideal $P=Rp$, and
$M$ is a pseudo-absorbing primary multiplication $R$-module. Then $M$ is a
direct sum of copies of $R/P^{n}$ $(n\geq 1)$. Moreover, every pseudo-
absorbing primary multiplication $R$-module apart from $R$ is pure-injective.

###### Proof.

Suppose that $R$ is a discrete valuation domain with maximal ideal $P=Rp$, and
$M$ is a pseudo-absorbing primary multiplication $R$-module. Suppose also that
$N$ is the indecomposable summand of $M$. It follows from Proposition 2.4 (2)
that $N$ is an indecomposable pseudo-absorbing primary multiplication module,
and so $N\cong R/P^{n}$ $(n\geq 1)$, by Theorem 2.6. For the“Moreover”
statement, since $M$ is a direct sum of copies of $R/P^{n}$, every pseudo-
absorbing primary multiplication $R$-module apart from $R$ is pure-injective,
by Theorem 2.6 and [8, Proposition 1.3].  $\Box$

##  3 The separated pseudo-absorbing primary multiplication modules over a
pullback ring

In this section, the indecomposable separated pseudo-absorbing primary
multiplication modules over a pullback of two local Dedekind domains are
described and some of their basic properties are given. Recall the
construction: if $R_{1}$ and $R_{2}$ are two discrete valuation domains with
maximal ideals $P_{1}=Rp_{1}$ and $P_{2}=Rp_{2}$ respectively, then

$R=(R_{1}\xrightarrow{\mathcal{V}_{1}(x)}\overline{R}\xleftarrow{\mathcal{V}_{2}(x)}R_{2})$
(1)

is the pullback ring of $R_{1}$ and $R_{2}$ with a unique maximal ideal
$P=P_{1}\oplus P_{2}$. Note that $\overline{R}\cong R/P\cong R_{1}/P_{1}\cong
R_{2}/P_{2}$ is a field. This means that $R$ is a Noetherian local ring with
unique maximal ideal $P$. So it is easy to see that $P_{1}$ and $P_{2}$ are
the other prime ideals of $R$. For more details on the pullback-theoretic of
two discrete valuation domains, see [8].

In the following results of this paper we assume that $R$ is a pullback ring
as in (1) and
$S=(S_{1}=S/P_{2}S\xrightarrow{f_{1}(x)}\overline{S}=S/PS\xleftarrow{f_{2}(x)}S_{2}=S/P_{1}S)$
is a separated $R$-module.

###### Proposition 3.1

Let the situation be as in (1). Then the following statements hold:

$(1)$ The ideals $\\{0\\}$, $0\oplus P_{2}^{m}$, $P_{1}^{n}\oplus 0$ and
$P_{1}^{n}\oplus P_{2}^{m}$ are $2$-absorbing primary ideals of $R$ for every
$m,n\geq 1$.

$(2)$ Suppose that $T=(T_{1},T_{2})$ is a pseudo-absorbing primary submodule
of the non-zero separated $R$-module
$S=(S_{1}=S/P_{2}S\xrightarrow{f_{1}(x)}\overline{S}=S/PS\xleftarrow{f_{2}(x)}S_{2}=S/P_{1}S)$.
Then $T_{1}$ and $T_{2}$ are pseudo-absorbing primary submodules of $S_{1}$
and $S_{2}$, respectively.

$(3)$ Suppose that $T=(T_{1},T_{2})$ is a pseudo-absorbing primary submodule
of the non-zero separated $R$-module
$S=(S_{1}=S/P_{2}S\xrightarrow{f_{1}(x)}\overline{S}=S/PS\xleftarrow{f_{2}(x)}S_{2}=S/P_{1}S)$.
Then $(T:_{R}S)\in\\{0,0\oplus P_{2}^{m},P_{1}^{n}\oplus 0,P_{1}^{n}\oplus
P_{2}^{m}\\}$ for some $m,n\geq 1$.

$(4)$ Suppose that $T=(T_{1},T_{2})$ is a pseudo-absorbing primary submodule
of the non-zero separated $R$-module
$S=(S_{1}=S/P_{2}S\xrightarrow{f_{1}(x)}\overline{S}=S/PS\xleftarrow{f_{2}(x)}S_{2}=S/P_{1}S)$
such that $S_{1}$ and $S_{2}$ are pseudo-absorbing primary multiplication
$R_{1}$-module and $R_{2}$-module, respectively. Then $(T:_{R}S)=0$ implies
$T=0$ and $S_{i}\neq 0$, for each $i=1,2$.

$(5)$ Let
$S=(S_{1}=S/P_{2}S\xrightarrow{f_{1}(x)}\overline{S}=S/PS\xleftarrow{f_{2}(x)}S_{2}=S/P_{1}S)$
be a non-zero separated pseudo-absorbing primary multiplication $R$-module.
Then $S$ contains at least one pseudo-absorbing primary submodule.

###### Proof.

$(1)$ It follows from [13, Proposition 3.1] that $\\{0\\}$ is a $2$-absorbing
primary ideal of $R$. Since $\sqrt{0\oplus P_{2}^{m}}=0\oplus P_{2}$,
$\sqrt{P_{1}^{n}\oplus 0}=P_{1}\oplus 0$ and $\sqrt{P_{1}^{n}\oplus
P_{2}^{m}}=P_{1}\oplus P_{2}$ are prime ideals of $R$, we conclude by [2,
Theorem 2.8] that $0\oplus P_{2}^{m}$, $P_{1}^{n}\oplus 0$ and
$P_{1}^{n}\oplus P_{2}^{m}$ are $2$-absorbing primary ideals of $R$ for every
$m,n\geq 1$.

$(2)$ Suppose that $T=(T_{1},T_{2})$ is a pseudo-absorbing primary submodule
of the non-zero separated $R$-module
$S=(S_{1}=S/P_{2}S\xrightarrow{f_{1}(x)}\overline{S}=S/PS\xleftarrow{f_{2}(x)}S_{2}=S/P_{1}S)$.
Suppose also that $a,b,c\in R_{1}$ and $abc\in(T_{1}:_{R_{1}}S_{1})$. Without
loss of generality we may assume that $a,b$ and $c$ are nonunit elements. Then
$a,b,c\in P_{1}$, and so $\mathcal{V}_{1}(abc)=0=\mathcal{V}_{2}(0)$ implies
$(abc,0)\in R$. We show that $(abc,0)\in(T:_{R}S)$. Suppose that
$(s_{1},s_{2})\in S$. Then $(abcs_{1},0)\in T$, since $abcs_{1}\in T_{1}\cap
P_{1}S$, $0\in T_{2}\cap P_{2}S$ and $f_{1}(abcs_{1})=0=f_{2}(0)$. Hence
$(abcs_{1},0)=(a,0)(b,0)(c,0)(s_{1},s_{2})\in T$. Thus $(abc,0)\in(T:_{R}S)$.
But $(T:_{R}S)$ is a $2$-absorbing primary ideal of $R$, hence either
$(a,0)(b,0)\in(T:_{R}S)$ or $(a,0)(c,0)\in\sqrt{(T:_{R}S)}$ or
$(b,0)(c,0)\in\sqrt{(T:_{R}S)}$. Therefore either $ab\in(T_{1}:_{R_{1}}S_{1})$
or $ac\in\sqrt{(T_{1}:_{R_{1}}S_{1})}$ or $bc\in\sqrt{(T_{1}:_{R_{1}}S_{1})}$.
Thus $T_{1}$ is a pseudo-absorbing primary submodule of $S_{1}$. A similar
argument shows that $T_{2}$ is a pseudo-absorbing primary submodule of
$S_{2}$.

$(3)$ Suppose that $T=(T_{1},T_{2})$ is a pseudo-absorbing primary submodule
of the non-zero separated $R$-module
$S=(S_{1}=S/P_{2}S\xrightarrow{f_{1}(x)}\overline{S}=S/PS\xleftarrow{f_{2}(x)}S_{2}=S/P_{1}S)$.
It follows from [2, Theorem 2.11] that $2$-absorbing primary ideals of $R_{1}$
are of the form $\\{0\\}$ and $P_{1}^{n}$ for some positive integer $n$ and
$2$-absorbing primary ideals of $R_{2}$ are of the form $\\{0\\}$ and
$P_{2}^{m}$ for some positive integer $m$. Hence
$(T:_{R}S)=(T_{1}:_{R_{1}}S_{1})\times(T_{2}:_{R_{2}}S_{2})$ gives
$(T:_{R}S)=0$, $0\oplus P_{2}^{m}$, $P_{1}^{n}\oplus 0$ or $P_{1}^{n}\oplus
P_{2}^{m}$ for some $m,n\geq 1$.

$(4)$ Suppose that $T=(T_{1},T_{2})$ is a pseudo-absorbing primary submodule
of the non-zero separated $R$-module
$S=(S_{1}=S/P_{2}S\xrightarrow{f_{1}(x)}\overline{S}=S/PS\xleftarrow{f_{2}(x)}S_{2}=S/P_{1}S)$
such that $S_{1}$ and $S_{2}$ are pseudo-absorbing primary multiplication
$R_{1}$-module and $R_{2}$-module, respectively. Since $T_{1}$ and $T_{2}$ are
pseudo-absorbing primary submodules of $S_{1}$ and $S_{2}$, respectively by
part $(2)$, we have $T_{1}=(T_{1}:_{R_{1}}S_{1})S_{1}$ and
$T_{2}=(T_{2}:_{R_{2}}S_{2})S_{2}$, and so it is enough to show that
$(T_{1}:_{R_{1}}S_{1})$ and $(T_{2}:_{R_{2}}S_{2})$ are zero. Suppose to the
contrary, $(T_{1}:_{R_{1}}S_{1})\neq 0$ and since $S$ is nonzero, we may
assume that $S_{1}\neq 0$. Hence there exists a nonzero element $r_{1}$ of
$(T_{1}:_{R_{1}}S_{1})\subseteq P_{1}$, and so
$\mathcal{V}_{1}(r_{1})=0=\mathcal{V}_{2}(0)$ implies $(r_{1},0)\in R$.
Therefore $(r_{1},0)(s_{1},s_{2})=(r_{1}s_{1},0)\in T$ for every
$(s_{1},s_{2})\in S$, because $r_{1}s_{1}\in T_{1}\cap P_{1}S$, $0\in
T_{2}\cap P_{2}S$, $f_{1}(r_{1}s_{1})=0=f_{2}(0)$ and $T$ is separated. Thus
$0\neq(r_{1},0)S\subseteq T$, a contradiction. We show that $S_{2}\neq 0$, for
otherwise it would follow that $S=(P_{1}\oplus 0)S$, and so $(0\oplus
P_{2})S=(0\oplus P_{2})(P_{1}\oplus 0)S=0$. This means that $0\neq(0\oplus
P_{2})\subseteq(0:_{R}S)\subseteq(T:_{R}S)$, a contradiction. Therefore, by a
similar argument as above we can see that $(T_{2}:_{R_{2}}S_{2})=0$. Thus
$T_{1}=T_{2}=0$ and $S_{i}\neq 0$, for each $i=1,2$.

$(5)$ Let
$S=(S_{1}=S/P_{2}S\xrightarrow{f_{1}(x)}\overline{S}=S/PS\xleftarrow{f_{2}(x)}S_{2}=S/P_{1}S)$
be a non-zero separated pseudo-absorbing primary multiplication $R$-module. It
follows from Theorem 3.2 that $S_{1}$ is a pseudo-absorbing primary
multiplication $R_{1}$-module. We show that there exists a pseudo-absorbing
primary submodule of $S_{1}$. Suppose to the contrary, there is no pseudo-
absorbing primary submodule of $S_{1}$. Since every prime submodule is a
pseudo-absorbing primary submodule of $S_{1}$, it can be seen by [23, Lemma
1.3, Proposition 1.4] that $S_{1}$ is a torsion divisible $R_{1}$-module with
$P_{1}S_{1}=S_{1}$ and $S_{1}$ is not finitely generated. By a similar
argument as in [9, Proposition 2.7 Case 2], we get $S_{1}\cong
E(R_{1}/P_{1})$, a contradiction, by Proposition 2.5. Therefore there exists a
pseudo-absorbing submodule $T_{1}$ of $S_{1}$. Now, let $g$ be the projection
map of $R$ onto $R_{1}$ and apply Proposotion 2.4 to this particular choice
and obtain that $T_{1}$ is a pseudo-absorbing primary $R$-submodule of
$S_{1}=S/P_{2}S$. Thus there exists at least one pseudo-absorbing primary
submodule of $S$, by Proposition 2.3 $(1)$.  $\Box$

###### Theorem 3.2

Let
$S=(S_{1}=S/P_{2}S\xrightarrow{f_{1}(x)}\overline{S}=S/PS\xleftarrow{f_{2}(x)}S_{2}=S/P_{1}S)$
be a non-zero separated $R$-module. Then $S$ is a pseudo-absorbing primary
multiplication $R$-module if and only if $S_{1}$ and $S_{2}$ are pseudo-
absorbing primary multiplication $R_{1}$-module and $R_{2}$-module,
respectively.

###### Proof.

First suppose that $S$ is a separated pseudo-absorbing primary multiplication
$R$-module. If $\overline{S}=0$, then [8, Lemma 2.7 (i)] gives $S=S_{1}\oplus
S_{2}$. Thus $S_{1}$ and $S_{2}$ are pseudo-absorbing primary multiplication
$R_{1}$-module and $R_{2}$-module, respectively, by Proposition 2.4 $(2)$. If
$\overline{S}\neq 0$, again by Proposition 2.4 $(1)$, we have $S_{1}\cong
S/(0\oplus P_{2})S$ is a pseudo-absorbing primary multiplication $R/(0\oplus
P_{2})\cong R_{1}$-module, since $(0\oplus P_{2})\subseteq((0\oplus
P_{2})S:S)$. By a similar argument, one may see that $S_{2}$ is a pseudo-
absorbing primary multiplication $R/(P_{1}\oplus 0)\cong R_{2}$-module. Now,
assume that $S_{1}$ and $S_{2}$ are pseudo-absorbing primary multiplication
$R_{1}$-module and $R_{2}$-module, respectively. Suppose that $T$ is a pseudo-
absorbing primary submodule of $S$. It follows from Proposition 3.1 $(3)$ that
$(T:_{R}S)=0$ or $0\oplus P_{2}^{m}$ or $P_{1}^{n}\oplus 0$ or
$P_{1}^{n}\oplus P_{2}^{m}$ for some $m,n\geq 1$. We distinguish three cases:
Case one: If $(T:_{R}S)=0$, then $T=(T:_{R}S)S=0$, by Proposition 3.1 $(3)$,
and so $S$ is a pseudo-absorbing primary multiplication $R$-module.
Case two: If $(T:_{R}S)=P_{1}^{n}\oplus P_{2}^{m}$ for some $m,n\geq 1$, then
[10, Proposition 4.2 (i)] gives $(T_{1}:_{R_{1}}S_{1})=P_{1}^{n}$ and
$(T_{2}:_{R_{2}}S_{2})=P_{2}^{m}$. Now apply Proposition 3.1 $(2)$ to obtain
$T_{1}=(T_{1}:_{R_{1}}S_{1})S_{1}=P_{1}^{n}S_{1}$ and
$T_{2}=(T_{2}:_{R_{2}}S_{2})S_{2}=P_{2}^{m}S_{2}$. To complete the proof, it
is only necessary to observe that $T=(P_{1}^{n}\oplus P_{2}^{m})S$. Clearly
$(P_{1}^{n}\oplus P_{2}^{m})S\subseteq T$. Suppose that $t=(t_{1},t_{2})\in
T$. Then there are $s_{1}\in S_{1}$ and $s_{2}\in S_{2}$ such that
$t_{1}=p_{1}^{n}s_{1}$ and $t_{2}=p_{2}^{m}s_{2}$. But $s_{1}\in S_{1}$ and
$s_{2}\in S_{2}$, hence there are $u_{1}\in S_{1}$ and $u_{2}\in S2$ such that
$(s_{1},u_{2}),(u_{1},s_{2})\in S$. This means that
$t=(p_{1}^{n}s_{1},p_{2}^{m}s_{2})=(p_{1}^{n},0)(s_{1},u_{2})+(0,p_{2}^{m})(u_{1},s_{2})\in(P_{1}^{n}\oplus
P^{m}_{2})S$, and thus $T=(P_{1}^{n}\oplus P^{m}_{2})S$.
Case three: If $(T:_{R}S)=P_{1}^{n}\oplus 0$ for some $n\geq 1$, then [10,
Proposition 4.2 (i)] gives $(T_{1}:_{R_{1}}S_{1})=P_{1}^{n}$ and
$(T_{2}:_{R_{2}}S_{2})=0$. Now apply Proposition 3.1 $(2)$ to obtain
$T_{1}=(T_{1}:_{R_{1}}S_{1})S_{1}=P_{1}^{n}S_{1}$,
$T_{2}=(T_{2}:_{R_{2}}S_{2})S_{2}=0$. To complete the proof, it is only
necessary to observe that $T=(P_{1}^{n}\oplus 0)S$. Clearly $(P_{1}^{n}\oplus
0)S\subseteq T$. Suppose that $t=(t_{1},0)\in T$. Then there are $s_{1}\in
S_{1}$ and $s_{2}\in S_{2}$ such that $t_{1}=p_{1}^{n}s_{1}$ and
$(s_{1},s_{2})\in S$. Hence
$t=(p_{1}^{n}s_{1},0)=(p_{1}^{n},0)(s_{1},s_{2})\in(P_{1}^{n}\oplus 0)S$, and
thus $T=(P_{1}^{n}\oplus 0)S$. By a similar argument as above, we can see that
$T=(0\oplus P_{2}^{m})S$ if $(T:_{R}S)=0\oplus P_{2}^{m}$ for some $m\geq 1$.
Therefore $S$ is a pseudo-absorbing primary multiplication $R$-module.

$\Box$

Consequently, in view of Theorems 2.6 and 3.2, we state the following
corollary.

###### Corollary 3.3

Suppose that $R$ is a pullback ring as described in (1). Then the following
separated $R$-modules are indecomposable and pseudo-absorbing primary
multiplication modules:

$(1)$ $R$;

$(2)$
$S=(S_{1}=R_{1}/P_{1}^{n}\xrightarrow{f_{1}(x)}\overline{R}\xleftarrow{f_{2}(x)}R_{2}/P_{2}^{m})$;

$(3)$
$S=(S_{1}=R_{1}\xrightarrow{f_{1}(x)}\overline{R}\xleftarrow{f_{2}(x)}R_{2}/P_{2}^{m})$;

$(4)$
$S=(S_{1}=R_{1}/P_{1}^{n}\xrightarrow{f_{1}(x)}\overline{R}\xleftarrow{f_{2}(x)}R_{2})$;

for every $m,n\geq 1$.

###### Proof.

It follows from Theorems 2.6 and 3.2 that these $R$-modules are pseudo-
absorbing primary multiplication modules and from [8, Lemma 2.8] that they are
indecomposable.  $\Box$

The observation in Corollary 3.3 provides a clue to a characterization of non-
zero indecomposable separated pseudo-absorbing primary multiplication
$R$-modules.

###### Theorem 3.4

Suppose that $R$ is a pullback ring as described in (1). Then every non-zero
indecomposable separated pseudo-absorbing primary multiplication $R$-module is
isomorphic to one of the modules listed in Corollary 3.3.

###### Proof.

Suppose that
$S=(S_{1}=S/P_{2}S\xrightarrow{f_{1}(x)}\overline{S}=S/PS\xleftarrow{f_{2}(x)}S_{2}=S/P_{1}S)$
is a non-zero indecomposable separated pseudo-absorbing primary multiplication
$R$-module such that $S\neq R$. It follows from Proposition 3.1 $(5)$ that
there exists at least one pseudo-absorbing primary submodule of $S$, say
$T=(T_{1},T_{2})$, and so $T_{i}$ is a pseudo- absorbing submodule $S_{i}$, by
Proposition 3.1 $(2)$. If $T_{i}=S_{i}$ for some $i\in\\{1,2\\}$, say
$T_{1}=S_{1}$, then $(T_{1}:_{R_{1}}S_{1})=R_{1}$ is a pseudo-absorbing
primary ideal of $R_{1}$, which contradicts [2, Theorem 2.11]. First we show
that $S\neq PS$. As remarked in Proposition 3.1 $(3)$, consider various cases
for $(T:_{R}S)$. Case one: If $(T:_{R}S)=0$, then $S\neq PS$ and [8, Lemma 2.7
(i)] implies $S=S_{1}\oplus S_{2}$. Since $S$ is indecomposable, we conclude
that either $S_{1}=0$ or $S_{2}=0$, and this contradicts Proposition 3.1
$(4)$. Case two: If $(T:_{R}S)=P_{1}^{n}\oplus P_{2}^{m}$ for some $m,n\geq
1$, then $T=(T:_{R}S)S=(P_{1}^{n}\oplus P_{2}^{m})S$, since $S$ is a pseudo-
absorbing primary multiplication $R$-module. This means that
$T_{1}=(T_{1}:_{R_{1}}S_{1})S_{1}=P_{1}^{n}S_{1}\subseteq
P_{1}S_{1}=PS/P_{2}S\neq S/P_{2}S=S_{1}$, and thus $PS\neq S$. Case three: If
$(T:_{R}S)=P_{1}^{n}\oplus 0$ for some $n\geq 1$, then
$T=(T:_{R}S)S=(P_{1}^{n}\oplus 0)S$, as $S$ is a pseudo-absorbing primary
multiplication $R$-module. This means that
$T_{1}=(T_{1}:_{R_{1}}S_{1})S_{1}=P_{1}^{n}S_{1}\subseteq
P_{1}S_{1}=PS/P_{2}S\neq S/P_{2}S=S_{1}$, and thus $PS\neq S$. Similarly, if
$(T:_{R}S)=0\oplus P_{2}^{m}$ for some $m\geq 1$, we get $PS\neq S$. Assume
that $s\in S_{1}\cup S_{2}$ such that $s\notin PS$ and

$o(s)=$ $\begin{cases}{\rm least}$ $m$ ${\rm such}$ ${\rm that}$ $P^{m}s$
${\rm=}$ $0&\text{if there is such $m$},\\\ \infty&\text{if there is no such
$m$}.\end{cases}$

We can assume that $s\in S_{2}$, and hence there exists $s_{1}\in S_{1}$ such
that $(s_{1},s=:s_{2})\in S$, $f_{1}(s_{1})=f_{1}(s_{2})$ and $o(s_{1})=n$ is
minimal. By a similar rgument as in [8, Theorem 2.9], we can see that
$R_{i}s_{i}$ is pure in $S_{i}$ for $i=1,2$. Thus, $R_{1}s_{1}\cong
R_{1}/P_{1}^{n}$ and $R_{2}s_{2}\cong R_{2}/P_{2}^{m}$ which are direct
summands in $S_{1}$ and $S_{2}$, respectively. If $o(s_{1})=\infty$ and
$o(s_{2})=\infty$, again by [8, Theorem 2.9], $R_{1}s_{1}\cong R_{1}$ and
$R_{2}s_{2}\cong R_{2}$ which are pure of $S_{1}$ and $S_{2}$, respectively.
Now suppose that $\overline{M}$ is the $\overline{R}$-subspace of
$\overline{S}$ which is generated by $\overline{s}=f_{1}(s_{1})=f_{1}(s_{2})$.
Then $\overline{M}\cong\overline{R}$. So that, on taking
$M=(R_{1}s_{1}=M_{1}\rightarrow\overline{M}\leftarrow M_{2}=R_{2}s_{2})$, we
deduce that $M$ is a pseudo-absorbing primary multiplication $R$-submodule of
$S$, by Corollary 3.3 which is a direct summand of $S$, and since $S$ is
indecomposable, we get $S=M$. Thus $S$ is one of the modules listed
$(2)$–$(4)$ in the Corollary 3.3 and this completes the proof.  $\Box$

In view of Theorem 3.4, we have the following result.

###### Corollary 3.5

Suppose that $R$ is a pullback ring as described in (1). Then every non-zero
indecomposable separated pseudo-absorbing primary multiplication $R$-module
non isomorphic to $R$ is pure-injective. Furthermore, every non-zero
indecomposable separated pseudo-absorbing primary multiplication $R$-module
non isomorphic to $R$ has a finite-dimensional top.

###### Proof.

It follows from Theorem 3.4 and [8, Theorem 2.9] that every non-zero
indecomposable separated pseudo-absorbing primary multiplication $R$-module
non isomorphic to $R$ is pure-injective. For the“Furthermore” statement,
suppose that
$S=(S_{1}=S/P_{2}S\xrightarrow{f_{1}(x)}\overline{S}=S/PS\xleftarrow{f_{2}(x)}S_{2}=S/P_{1}S)$
is a non-zero indecomposable separated pseudo-absorbing primary multiplication
$R$-module such that $S\neq R$. Then $S_{1}$ and $S_{2}$ are pseudo-absorbing
primary multiplication $R_{1}$-module and $R_{2}$-module, respectively, by
Theorem 3.2. Suppose also that $T(S_{i})$ is the torsion submodule of $S_{i}$
such that $T(S_{i})\neq S_{i}$ for $i=1,2$. Then $T(S_{i})$ is a prime
submodule of $S_{i}$ for $i=1,2$, and so is a pseudo-absorbing primary
submodule of $S_{i}$ such that $(T(S_{i}):_{R_{i}}S_{i})=0$ for $i=1,2$ (see
[24, Lemma 3.8]). Thus $T(S_{i})=0$ for each $i=1,2$, and hence $S_{i}$ is
either torsion or torsion-free for each $i=1,2$. Now apply Theorem 3.4 and
[11, Theorem 3.14] to see that every non-zero indecomposable separated pseudo-
absorbing primary multiplication $R$-module non isomorphic to $R$ has a
finite-dimensional top.  $\Box$

##  4 The non-separated pseudo-absorbing primary multiplication modules over a
pullback ring

Let the situation be as in (1), so that $R$ is a pullback ring as in (1) and
$S=(S_{1}=S/P_{2}S\xrightarrow{f_{1}(x)}\overline{S}=S/PS\xleftarrow{f_{2}(x)}S_{2}=S/P_{1}S)$
is a separated $R$-module. In this section, the indecomposable non-separated
pseudo-absorbing primary multiplication modules with finite-dimensional top
over a pullback of two local Dedekind domains are described. In fact, it turns
out that non-separated pseudo-absorbing primary multiplication modules with
finite-dimensional top can be obtained by amalgamating finitely many separated
indecomposable pseudo-absorbing primary multiplication modules.

Suppose that $M$ is an indecomposable non-separated pseudo-absorbing primary
multiplication $R$-module with finite-dimensional top and the exact sequence
$0\rightarrow K\xrightarrow{i}S\xrightarrow{\varphi}M\rightarrow 0$ is a
separated representation of $M$. We show that $S$ is a direct sum of only
finitely many indecomposable pseudo-absorbing primary multiplication modules.
First we show that $M$ is a pseudo-absorbing primary multiplication $R$-module
if and only if $S$ is a pseudo-absorbing primary multiplication $R$-module.

###### Proposition 4.1

Suppose that $R$ is a pullback ring as described in (1) and $0\rightarrow
K\xrightarrow{i}S\xrightarrow{\varphi}M\rightarrow 0$ is a separated
representation of $M$. If $M$ is non-separated, then
$(T:_{R}S)\notin\\{0,P_{1}^{n}\oplus 0,0\oplus P_{2}^{m}\\}$ for each
submodule $T$ of $S$ and $m,n\geq 1$.

###### Proof.

Let $T$ be a submodule of $S$ such that $(T:_{R}S)\in\\{0,P_{1}^{n}\oplus
0,0\oplus P_{2}^{m}\\}$ for some $m,n\geq 1$. First, suppose that
$(T:_{R}S)=0$, and so $(0:_{R}S)=0$. Let $r=(r_{1},r_{2})\in(0:_{R}M)$. Then
$rM=rS/K=0$, which means that $rS\subseteq K$. Hence $rPS\subseteq PK=0$, by
[20, Proposition 2.4], and so $rP=0$. Therefore $r_{1}p_{1}=0=r_{2}p_{2}$;
thus $r=(0,0)$, and so $(0:_{R}M)=0$. This implies that $(P_{1}\oplus
0)\nsubseteq(0:_{R}M)$ and $(0\oplus P_{2})\nsubseteq(0:_{R}M)$. Without loss
of generality, assume that $(P_{1}\oplus 0)M\cap(0\oplus P_{2})M\neq 0$; hence
there exists a non-zero $m\in(P_{1}\oplus 0)M\cap(0\oplus P_{2})M$. But
$\varphi$ is an $R$-homomorphism of $S$ onto $M$, then there exists $x\in S$
such that $m=\varphi(x)$. Since
$\varphi^{-1}(Rm)=\varphi^{-1}(\varphi(Rx))=Rx$ and $Rm\neq 0$, it follows
from [13, Lemma 4.1] that $0\rightarrow K\rightarrow Rx\rightarrow
Rm\rightarrow 0$ is a separated representation of $Rm$ such that $K\subseteq
PRx$. Therefore, there exist $m_{1},m_{2}\in M$ and $s,t\in\mathbb{N}$ such
that $m=(p_{1}^{s},0)m_{1}=(0,p_{2}^{t})m_{2}$. It follows that
$Pm=(P_{1}\oplus P_{2})m=0$, and hence $\varphi(Px)=0$. Thus
$\varphi(P_{1}x)=\varphi(P_{2}x)=0$. Since $\varphi$ is one-to-one on $P_{1}S$
and on $P_{1}S$, we conclude that $Px=0$. Therefore $K\subseteq PRx=0$, and
hence $M\cong S$ is a separated $R$-module which is a contradiction. So
suppose that $(T:_{R}S)=P_{1}^{n}\oplus 0$ for some $n\geq 1$. Then
$(0:_{R}S)\subseteq(T:_{R}S)$ implies that either $(0:_{R}S)=0$ or
$(0:_{R}S)=P_{1}^{n_{1}}\oplus 0$ for some $n_{1}\geq n$. The first case
follows from the above comment. Assume that $(0:_{R}S)=P_{1}^{n_{1}}\oplus 0$
for some $n_{1}\geq n$. Then $(P_{1}^{n_{1}}\oplus 0)S=0$, which implies that
$P_{1}^{n_{1}}\oplus 0\subseteq((0\oplus P_{2})S:_{R}S)$. Since $0\oplus
P_{2}\subseteq((0\oplus P_{2})S:_{R}S)$, we conclude that $P_{1}^{n_{1}}\oplus
P_{2}\subseteq((0\oplus P_{2})S:_{R}S)$. Therefore, by [13, Proposition 4.2],
we get $K\subseteq P^{n_{1}}S\subseteq(P_{1}^{n_{1}}\oplus
P_{2})S\subseteq(0\oplus P_{2})S$. Thus $K=0$ because $K\cap(0\oplus
P_{2})S=0$, by [20, Proposition 2.3], and hence $M\cong S$ is a separated
$R$-module which is a contradiction. A similar argument shows that
$(T:_{R}S)\neq 0\oplus P_{2}^{m}$ for every $m\geq 1$, and the proof is
complete.  $\Box$

###### Theorem 4.2

Suppose that $R$ is a pullback ring as described in (1) and $0\rightarrow
K\xrightarrow{i}S\xrightarrow{\varphi}M\rightarrow 0$ is a separated
representation of $M$ such that $M$ is non-separated $R$-module. Then $M$ is a
pseudo-absorbing primary multiplication $R$-module if and only if $S$ is a
pseudo-absorbing primary multiplication $R$-module.

###### Proof.

First, suppose that $S$ is a pseudo-absorbing primary multiplication
$R$-module. Since $S\cong M/K$, the proof follows from Proposition 2.3 $(1)$.
We now turn to the converse statement, which is the non-trivial part of the
result. Let $M$ be a pseudo-absorbing primary multiplication $R$-module and
$T$ a pseudo-absorbing primary submodule of $S$. It follows from Proposition
4.1 that $(T:_{R}S)\notin\\{0,P_{1}^{n}\oplus 0,0\oplus P_{2}^{m}\\}$ for each
submodule $T$ of $S$ and $m,n\geq 1$, as $M$ is non-separated, and so
Proposition 3.1 $(3)$ gives $(T:_{R}S)=P_{1}^{n}\oplus P_{2}^{m}$ for some
$m,n\geq 1$. Assume that $n\geq m$. Therefore, by [13, Proposition 4.2], we
get $K\subseteq P^{n}S\subseteq(P_{1}^{n}\oplus P_{2}^{m})S\subseteq T$. By
using the fact that $T/K$ is a pseudo-absorbing primary submodule of $S/K\cong
M$ and $M$ is pseudo-absorbing primary multiplication, we can see that
$T/K=(T/K:_{R}S/K)S/K=(P_{1}^{n}\oplus P_{2}^{m})(S/K)=((P_{1}^{n}\oplus
P_{2}^{m})S+K)/K=((P_{1}^{n}\oplus P_{2}^{m})S)/K$, and thus
$T=(P_{1}^{n}\oplus P_{2}^{m})S$. Hence $S$ is a pseudo-absorbing primary
multiplication $R$-module.  $\Box$

###### Corollary 4.3

Suppose that $R$ is a pullback ring as described in (1) and $0\rightarrow
K\xrightarrow{i}S\xrightarrow{\varphi}M\rightarrow 0$ is a separated
representation of $M$ such that $M$ is an indecomposable non-separated pseudo-
absorbing primary multiplication $R$-module with finite-dimensional top over
$\overline{R}$. Then $S$ is a pure-injective $R$-module with finite-
dimensional top.

###### Proof.

By Theorem 4.2, $S$ is a pseudo-absorbing primary multiplication $R$-module.
Hence $S$ is a pure-injective $R$-module, by Corollary 3.5. It follows from
[8, Proposition 2.6 (i)] that $S/PS\cong M/PM$; thus $S$ is finite-dimensional
top.  $\Box$

Our next goal is to show that if $M$ is an indecomposable non-separated
pseudo-absorbing primary multiplication $R$-module with finite-dimensional top
and the exact sequence $0\rightarrow
K\xrightarrow{i}S\xrightarrow{\varphi}M\rightarrow 0$ is a separated
representation of $M$, then $S$ is a direct sum of only finitely many
indecomposable pseudo-absorbing primary multiplication modules. But it seems
that the solving of this problem depends on the results found in [8, Lemma
3.1, Proposition 3.2 and Proposition 3.4]. In fact, the proofs of these
results are perhaps needed more than the result itself. Note that the key
properties in those results are the pure-injectivity of $S$ which is obtained
from the pure-injectivity of $M$ (see [8, Proposition2.6 (ii)]), the
indecomposability and the nonseparability of $M$. But in this paper the pure-
injectivity of $S$ is obtained from the pseudo-absorbing primary multiplicity
of $M$ (see Corollary 4.3). Therefore, we can replace the statement “$M$ is
pure-injective” by “$M$ is pseudo-absorbing primary multiplication”. However,
there is one beautiful existence result concerning the direct summands of $S$.

###### Corollary 4.4

Suppose that $R$ is a pullback ring as described in (1) and $0\rightarrow
K\xrightarrow{i}S\xrightarrow{\varphi}M\rightarrow 0$ is a separated
representation of $M$ such that $M$ is an indecomposable non-separated pseudo-
absorbing primary multiplication $R$-module with finite-dimensional top over
$\overline{R}$. Then $S$ is a direct sum of finitely many indecomposable
pseudo-absorbing primary multiplication modules.

###### Proof.

First we show that $R$ does not occur among the direct summands of $S$. Assume
that $S=R\oplus N$ for some submodule $N$ of $S$. Then $K\subseteq N$, because
Soc$(R)=0$. Hence $M\cong N/K\oplus R$, a contradiction, since $M$ is
indecomposable and non-separated. Therefore, in view of the observations in
the above paragraph, $S$ is a direct sum of finitely many indecomposable
pseudo-absorbing primary multiplication modules (described in $(2)$–$(4)$ of
Corollary 3.3), by Proposition 2.4 $(2)$ and Theorem 4.2.  $\Box$

###### Corollary 4.5

Let the situation be as in Corollary 4.4. Then the following statements hold:

$(1)$ The quotient fields $Q(R_{1})$, $Q(R_{2})$ do not occur among the direct
summands of $S$.

$(2)$ At most two copies of modules of infinite length can occur among the
indecomposable summands of $S$.

We are now ready for the main result of this paper. But before starting let us
quickly describe its idea. Let the situation be as in (1) and let
$0\rightarrow K\xrightarrow{i}S\xrightarrow{\varphi}M\rightarrow 0$ be a
separated representation of an $R$-module $M$. It follows from Corollary 4.4
that if $M$ is indecomposable non-separated pseudo-absorbing primary
multiplication with finite-dimensional top over $\overline{R}$, then $S$ is a
direct sum of finitely many indecomposable pseudo-absorbing primary
multiplication modules, and we have shown that these modules are as in
Corollary 3.3. It should be observed that $R$, $Q(R_{1})$ and $Q(R_{2})$ (the
quotient fields of $R_{1}$ and $R_{2}$) cannot be amalgamated with any other
direct summands of $S$, and hence cannot take place in any separated
representation $0\rightarrow
K\xrightarrow{i}S\xrightarrow{\varphi}M\rightarrow 0$ because
$(0:_{R}ker(\varphi))=(0:_{R}K)=P$ which means that $K\subseteq{\rm Soc}(S)$,
and so $M$ is obtained by amalgamation in the socles of the various direct
summands of $S$ whereas $0={\rm Soc}(R)={\rm Soc}(Q(R_{1}))={\rm
Soc}(Q(R_{2}))$ and this clarify the proofs of Corollies 4.4 and 4.5. Thus
natural questions of these sort are: do these conditions give us extra
conditions on the possible direct summands of $S$? How can we get the direct
summands of $M$ using the amalgamated summands of $S$? The goal of this paper
is to answer these questions in the case in which $M$ is indecomposable non-
separated pseudo-absorbing primary multiplication module with finite-
dimensional top. Previously, these questions are answered by Levy in [21, 22]
for the case of finitely generated $R$-modules $M$. We recall what we will
need from that. In fact, Levy showed that the indecomposable finitely
generated $R$-modules are of two non-overlapping types which he called them
deleted cycle and block cycle types. We observe that deleted cycle modules are
very pertinent to us. We get such modules from a direct summand, $S$, of
indecomposable separated modules by amalgamating the direct summands of $S$ in
pairs to form a chain but leaving the two ends unamalgamated. By considering
the fact that the dimension of the socle of any finitely generated
indecomposable separated module over $R$ is smaller than two, each
indecomposable summand of $S$ may be amalgamated with at most two other
indecomposable summands. We give an example of a non-separated module which is
obtained by amalgamating the summands of a separated $R$-module $S$:

$S_{1}=(R_{1}/P_{1}^{5}\rightarrow\overline{R}\leftarrow R_{2}/P_{2}^{3})=Ra$
with $P_{2}^{3}a=0=P_{1}^{5}a$

$S_{2}=(R_{1}/P_{1}^{4}\rightarrow\overline{R}\leftarrow
R_{2}/P_{2}^{2})=Ra^{\prime}$ with $P_{1}^{4}a^{\prime}=0=P_{2}^{2}a^{\prime}$

and so we can form the non-separated module $(S_{1}\oplus
S_{2})/(R(p_{2}^{2}a-p_{1}^{3}a^{\prime})=Rc+Rc^{\prime}$ such that
$c=a+R(p_{2}^{2}a-p_{1}^{3}a^{\prime})$,
$c^{\prime}=a^{\prime}+R(p_{2}^{2}a-p_{1}^{3}a^{\prime})$ and
$P_{2}^{3}c=0=P_{1}^{4}c^{\prime}=P_{2}^{2}c=P_{1}^{3}c^{\prime}$, which is
obtained by identifying the “$P_{2}$-part” of the socle of the separated
module $S_{1}$ with the “$P_{1}$-part” of the socle of the separated module
$S_{2}$. See the following schematic version of diagram as depicted in Figure
1:

$a$$p_{1}a$$p_{1}^{2}a$$p_{1}^{3}a$$p_{1}^{4}a$$0$$p_{2}a$$p_{2}^{2}a$$0$$p_{1}^{3}a^{\prime}$$p_{1}^{2}a^{\prime}$$p_{1}a^{\prime}$$0$$a^{\prime}$$p_{2}a^{\prime}$$0$

$0$$0$$0$

Figure 1: A schematic version of the diagrams of a non-separated module which
is obtained by identifying the “$P_{2}$-part” of the socle of the separated
module $S_{1}$ with the “$P_{1}$-part” of the socle of the separated module
$S_{2}$.

The non-zero indecomposable pseudo-absorbing primary multiplication non-
separated $R$-modules are obtained in just the same way as indecomposable
finitely generated ones; we will use the same detailing but with pseudo-
absorbing primary multiplication separated modules in place of the finitely
generated ones. We have seen in Corollary 4.5 that any non-zero indecomposable
separated pseudo-absorbing primary multiplication module with $1$-dimensional
socle may occur only at one of the ends of the amalgamation chain, and thus we
have to show that the modules obtained by these amalgamations are
indecomposable pseudo-absorbing primary multiplication. Hence, the
indecomposable pseudo-absorbing primary multiplication non-separated modules
with finite-dimensional top will be classified in the following result.

###### Theorem 4.6

Suppose that $R$ is a pullback ring as described in (1). Then the
indecomposable non-separated pseudo-absorbing primary multiplication
$R$-modules $M$ with finite-dimensional top over $\overline{R}$, up to
isomorphism, are the following.

$(1)$ The indecomposable modules of finite length (apart from $R/P$ since it
is a separated module), i.e, $M=\sum^{s}_{i=1}Ra_{i}$ such that
$p_{1}^{n_{s}}a_{s}=p_{2}^{m_{1}}a_{1}=0$,
$p_{1}^{n_{i}-1}a_{i}=p_{2}^{m_{i+1}-1}a_{i+1}$ where $(1\leq i\leq s-1)$ and
$m_{i},n_{i}\geq 2$ except the case $m_{1},n_{s}\geq 1$.

$(2)$ $M=R/P_{1}^{n}+\sum^{s-1}_{i=1}Ra_{i}$ such that
$p_{1}^{n_{s-1}}a_{s-1}=0$, $p_{1}^{n-1}a_{0}=p_{2}^{m_{1}-1}a_{1}$ and
$p_{1}^{n_{i}-1}a_{i}=p_{2}^{m_{i+1}-1}a_{i+1}$ where $R/P_{1}^{n}\cong
Ra_{0}$, $(1\leq i\leq s-2)$ and $n,m_{i},n_{i}\geq 2$ except the case
$n_{s-1}\geq 1$.

$(3)$ $M=\sum^{s-1}_{i=1}Ra_{i}+R/P_{2}^{n}$ such that
$p_{2}^{m_{s-1}}a_{s-1}=0$, $p_{2}^{n-1}b_{0}=p_{1}^{n_{1}-1}a_{1}$ and
$p_{2}^{m_{i}-1}a_{i}=p_{1}^{n_{i+1}-1}a_{i+1}$ where $R/P_{2}^{n}\cong
Rb_{0}$, $(1\leq i\leq s-2)$ and $n,m_{i},n_{i}\geq 2$ except the case
$m_{s-1}\geq 1$.

$(4)$ $M=R/P_{1}^{r}+\sum^{s-2}_{i=1}Ra_{i}+R/P_{2}^{n}$ such that
$p_{2}^{n-1}b_{0}=p_{1}^{n_{1}-1}a_{1}$,
$p_{1}^{r-1}a_{0}=p_{2}^{m_{s-2}-1}a_{s-2}$ and
$p_{2}^{m_{i}-1}a_{i}=p_{1}^{n_{i+1}-1}a_{i+1}$ where $R/P_{1}^{r}\cong
Ra_{0}$, $R/P_{2}^{n}\cong Rb_{0}$, $(1\leq i\leq s-3)$ and
$n,m_{i},n_{i},r\geq 2$ except the case $m_{1},n_{s}\geq 1$.

###### Proof.

Suppose that $R$ is a pullback ring as described in (1) and $0\rightarrow
K\xrightarrow{i}S\xrightarrow{\varphi}M\rightarrow 0$ is a separated
representation of $M$ such that $M$ is indecomposable non-separated pseudo-
absorbing primary multiplication $R$-module with finite-dimensional top over
$\overline{R}$. Then $S$ is a direct sum of finitely many indecomposable
pseudo-absorbing primary multiplication modules, by Corollary 4.4. But we have
seen already that every indecomposable non-separated pseudo-absorbing primary
multiplication $R$-module has one of these forms so it remains to show that
the modules obtained by these amalgamation are, indeed, indecomposable pseudo-
absorbing primary multiplication $R$-modules. The pseudo-absorbing primary
multiplicity of these modules follows from Proposition 2.3 $(1)$ and Theorem
4.2, and the indecomposability follows from [22, 1.9] and [8, Theorem 3.5].
$\Box$

We close this section with the following corollary.

###### Corollary 4.7

Suppose that $R$ is a pullback ring as described in (1). Then every
indecomposable pseudo-absorbing primary multiplication $R$-module with finite-
dimensional top is pure-injective.

###### Proof.

The proof follows from Corollary 4.3, Theorem 4.6 and [8, Theorem 3.5].
$\Box$

##  5 Conclusion

This research contributes to the idea of pseudo-absorbing primary
multiplication modules over the pullback ring
$R=(R_{1}\xrightarrow{\mathcal{V}_{1}(x)}\overline{R}\xleftarrow{\mathcal{V}_{2}(x)}R_{2})$
of two discrete valuation domains $R_{1},R_{2}$ with common factor field
$\overline{R}$ and gives a complete description of these modules. The main
results of this article include the classification of indecomposable pseudo-
absorbing primary multiplication modules over $k[x,y:xy=0]_{(x,y)}$ (a typical
example of a pullback ring of two discrete valuation domains) where $k$ is a
field with finite-dimensional top, which is divided into two steps:
indecomposable separated pseudo-absorbing primary multiplication and
indecomposable non-separated pseudo-absorbing primary multiplication
$R$-modules. In the future, this work will be expanded to explore the
indecomposable pseudo-absorbing primary multiplication modules with finite-
dimensional top over the pullback ring of two discrete valuation domains
$R_{1},R_{2}$ onto a semi-simple artinian ring $\overline{R}$ and also over
the pullback ring of two local rings $R_{1}$ and $R_{2}$ onto a field
$\overline{R}$.

## References

  * [1] I. Assem, D. Simson and A. Skowronski, Elements of the Representation Theory of Associative Algebras, Vol. 1. Techniques of Representation Theory, London Math. Soc., Student Texts 65, Cambridge Univ. Press, Cambridge, 2007.
  * [2] A. Badawi, U. Tekir and E. Yetkin, On $2$-absorbing primary ideals in commutative rings, Bull. Korean Math. Soc., 51(4) (2014), 1163–1173.
  * [3] H. Bass, On the ubiquity of Gorenstein rings, Math. Z., 82 (1963), 8–29.
  * [4] D. Bennis, B. Fahid, Rings in which every $2$-absorbing ideal is prime, Beitr Algebra Geom, 2017.
  * [5] W. Crawley-Boevey, Tame algebras and generic modules, Proc. London Math. Soc., 63 (1991), 241–265.
  * [6] Yu. A. Drozd, Matrix problems and categories of matrices, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov., (LOMI) 28 (1972), 144–153 (in Russian).
  * [7] S. Ebrahimi Atani, Indecomposable weak multiplication modules over Dedekind domains, Demonstratio Math., 41 (2008), 33–43.
  * [8] S. Ebrahimi Atani, On pure-injective modules over pullback rings, Comm. Algebra, 28 (2000), 4037–4069.
  * [9] S. Ebrahimi Atani, On secondary modules over pullback rings, Comm. Algebra, 30 (2002), 2675–2685.
  * [10] R. Ebrahimi Atani and S. Ebrahimi Atani, On semiprime multiplication modules over pullback rings, Comm. Algebra, 41 (2013), 776–791.
  * [11] S. Ebrahimi Atani and F. Esmaeili Khalil Saraei, On $P$-torsion modules over a pullback of two Dedekind domains, Georgian Math. J., 19 (2012), 473–488.
  * [12] F. Esmaeili Khalil Saraei, On pseudo-prime multiplication modules over pullback rings, Colloq. Math., 143 (2016), 63–78.
  * [13] S. Ebrahimi Atani, S. Dolati Pish Hesari, M. Khoramdel, M. Sedghi Shanbeh Bazari, Pseudo-absorbing multiplication modules over a pullback ring, J. Pure Appl. Algebra, 222, no. 10 (2018), 3124–3136.
  * [14] Z.A. El-Bast, P.F. Smith, Multiplication modules, Comm. Algebra, 16 (1988), 755–779.
  * [15] J. Haefner and L. Klingler, Special quasi-triads and integral group rings of finite representation type I, J. Algebra, 158 (1993), 279–322.
  * [16] J. Haefner and L. Klingler, Special quasi-triads and integral group rings of finite representation type II, J. Algebra, 158 (1993), 323–374.
  * [17] I. Kaplansky, Modules over Dedekind rings and valuation rings, Trans. Amer. Math. Soc., 72 (1952), 327–340.
  * [18] V. V. Kirichenko, Classification of the pairs of mutually annihilating operators in a graded space and representations of a dyad of generalized uniserial algebra, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov., (LOMI) 75 (1978), 91–109, 196–197 (in Russian).
  * [19] L. Klingler, Integral representation of groups of square-free order, J. Algebra, 129 (1990), 26–74.
  * [20] L. S. Levy, Modules over pullbacks and subdirect sums, J. Algebra, 71(1) (1981), 50–61.
  * [21] L. Levy, Modules over Dedekind-like rings, J. Algebra, 93 (1985), 1–116.
  * [22] L. Levy, Mixed modules over $\mathbb{Z}$G, G cyclic of prime order, and over related Dedekind pullbacks, J. Algebra, 71 (1981), 62–114.
  * [23] R. L. MacCasland, M. E. Moore and P. F. Smith, On the spectrum of a module over a commutative ring, Comm. Algebra, 25 (1997), 79–103.
  * [24] M. Moore and S. J. Smith, Prime and radical submodules of modules over commutative rings, Comm. Algebra, 10 (2002), 5037–5064.
  * [25] H. Mostafanasab, E. Yetkin, U. Tekir, and A. Y. Darani, On $2$-absorbing primary submodules of modules over commutative rings, An. St. Univ. Ovidius Constanta, 24 (2016), no. 1, 335–351.
  * [26] M. Prest, Model Theory and Modules, London Mathematical Society, Cambridge University Press, Cambridge, 1988.
  * [27] M. Prest, Ziegler spectra of tame hereditary algebras, J. Algebra, 207 (1998), 146–164.
  * [28] C. M. Ringel, The Ziegler spectrum of a tame hereditary algebra, Colloq. Math., 76 (1998), 105–115.
  * [29] D. Simson, On representation types of modules subcategories and orders, Bull. Polish Acad. Sci. Math., 41 (1993), 77–93.
  * [30] D. Simson, On Corner type endo-wild algebras, J. Pure Appl. Algebra, 202 (2005), 118–132.
  * [31] D. Simson and A. Skowronski, Elements of the Representation Theory of Associa- tive Algebras, Vol. 3. Representation-Infinite Tilted Algebras, London Math. Soc., Student Texts 72, Cambridge Univ. Press, Cambridge, 2007.
  * [32] A. N. Wiseman, Projective modules over pullback rings, Math. Proc. Cambridge Philos. Soc., 97 (1985), 399–406.