#  Rings with finite $n$-Weak injective dimension and $(n,k)$-Weak cotorsion
modules

Mostafa Amini1,a, Houda Amzil2,b and Driss Bennis2,c

1\. Department of Mathematics, Faculty of Sciences, Payame Noor University,
Tehran, Iran.
2\. Department of Mathematics, Faculty of Sciences, Mohammed V University in
Rabat, Morocco.
$\mathbf{a.}$<EMAIL_ADDRESS>
$\mathbf{b.}$<EMAIL_ADDRESS><EMAIL_ADDRESS>
$\mathbf{c.}$<EMAIL_ADDRESS><EMAIL_ADDRESS>

Abstract. Let $R$ be a ring and $n,k$ be two non-negative integers. As an
extension of several known notions, we introduce and study $(n,k)$-weak
cotorsion modules using the class of right $R$-modules with $n$-weak flat
dimensions at most $k$. Various examples and applications are also given.

Keywords: $(n,k)$-Weak cotorsion module; $n$-weak injective module; $n$-weak
flat module; $n$-super finitely presented module.

2010 Mathematics Subject Classification. 16D80, 16E05, 16E30, 16E65

##  1 Introduction

Throughout this paper, $R$ will denote an associative (not necessarily
commutative) ring with identity. As usual, we denote by $R$-${\rm Mod}$ and
${\rm Mod}$-$R$ the category of left $R$-modules and right R-modules,
respectively. The class of all right (resp. left) $R$-modules of flat
dimension at most a non-negative integer $k$ is denoted by
$\mathscr{F}_{k}(R^{op})$ (resp. $\mathscr{F}_{k}(R)$) and for a module $M$,
$E(M)$ denotes its injective envelope.

In 1984, Enochs introduced the concept of cotorsion modules as a
generalization of cotorsion abelian groups [6]. It differs from Matlis’
definition in [16] whose concern was with domains. Since then, the
investigation of these modules have become a vigorously active area of
research. We refer the reader to [2, 6, 15] for background on cotorsion
modules. In 2007, Mao and Ding in [14] introduced the concept of $k$-cotorsion
left modules as a new classification of cotorsion modules by using the class
$\mathscr{F}_{k}(R)$.

In 2015, Gao and Wang in [11] introduced the concept of weak injective and
weak flat modules by using super finitely presented modules instead of
finitely presented modules. Then, they investigated the properties of
$R$-modules with weak injective and weak flat dimension at most a non-negative
integer $k$. Also, Gao and Huang in [10], investigated the right
$\mathcal{WI}$-dimension of modules in terms of left derived functors of Hom
and the left $\mathcal{WI}$-resolutions of modules, where $\mathcal{WI}$ is
the class of weak injective $R$-modules. This process continues in 2018 when
Zhao in [24] introduced another relative derived functor in terms of left or
right $\mathcal{WF}$-resolutions to compute the weak flat dimension of
modules, where $\mathcal{WF}$ is the class of weak flat $R$-modules. Then,
they investigated homological properties of modules with finite weak injective
and weak flat dimensions. Let $\mathcal{WI}_{k}(R)$ and
$\mathcal{WF}_{k}(R^{op})$ be, respectively, the classes of left and right
$R$-modules with weak injective and weak flat dimensions at most $k$,
respectively. In 2018, Selvaraj and Prabakaran in [20], introduced a
particular case of $k$-cotorsion modules and called them $k$-weak cotorsion
modules by using the class $\mathcal{WF}_{k}(R^{op})$ instead of
$\mathscr{F}_{k}(R^{op})$.

Recently in [1], the notion of $n$-super finitely presented left $R$-modules
for any non-negative integer $n$ has been introduced as a particular case of
super finitely presented. Then, by using $n$-super finitely presented left
$R$-modules, a natural extension of the notions of weak injective and weak
flat modules have been introduced and studied. They are called the $n$-weak
injective and $n$-weak flat modules, respectively. Furthermore, some
homological aspects of modules with finite $n$-weak injective and $n$-weak
flat dimensions have been also investigated in [1]. Throughout,
$\mathcal{WI}^{n}(R)$, $\mathcal{WF}^{n}(R^{op})$, $\mathcal{WI}_{k}^{n}(R)$
and $\mathcal{WF}_{k}^{n}(R^{op})$ will denote respectively the classes of
$n$-weak injective left $R$-modules, $n$-weak flat right $R$-modules, left
$R$-modules with $n$-weak injective dimensions less than or equal to $k$ and
right $R$-modules with $n$-weak flat dimensions less than or equal to $k$.

In this paper, a natural extension of the notion of weak cotorsion modules is
introduced and studied which is called $(n,k)$-weak cotorsion modules (see
Definition 3.1). Every $(n,k)$-weak cotorsion module is $k$-weak cotorsion,
$k$-cotorsion and cotorsion, and $(0,k)$-weak cotorsion modules are exactly
$k$-weak cotorsion modules.

The paper is organized as follows:

In Sec. 2, some fundamental concepts and some preliminary results are stated.
Namely, we recall the definitions of $n$-super finitely presented, $n$-weak
injective left $R$-modules, and $n$-weak flat right $R$-modules introduced in
[1] (see Definitions 2.1 and 2.3).

In Sec. 3, we introduce $(n,k)$-weak cotorsion modules. Then, examples are
given to show that for $n>m\geq 0$, $(m,k)$-weak cotorsion $R$-modules need
not be $(n,k)$-weak cotorsion (see Remarks 3.2). Moreover, we show that if
$n$-${\rm wid}_{R}(R)\leq k$ and $M$ is an $R$-module, then $M$ is an
$(n,k)$-weak cotorsion if and only if $M$ is a direct sum of an injective
right $R$-module and a reduced $(n,k)$-weak cotorsion right $R$-module (see
Theorem 3.8).

In Sec. 4, we prove some equivalent characterizations of rings with finite
$n$-super finitely presented dimensions. Namely, for a coherent ring $R$ such
that $n$-${\rm wid}_{R}(_{R}R)\leq k$, we prove that ${\rm
l.n.sp.gldim}(R)\leq k$ if and only if every right $R$-module in
$\mathcal{WI}_{k-1}^{n}(R)^{\top}$ is $n$-weak flat if and only if every right
$R$-module in $\mathcal{WI}_{k}^{n}(R)^{\top}$ is $n$-weak flat if and only if
every left $R$-module in ${}^{\bot}\mathcal{WI}_{k}^{n}(R)$ belongs to
$\mathcal{WI}_{k}^{n}(R)$ if and only if every left $R$-module in
$\mathcal{WI}_{k}^{n}(R)^{\bot}$ is $n$-weak injective if and only if every
left $R$-module has a monic $\mathcal{WI}_{k-1}^{n}(R)$-cover (see Theorem
4.4). Then, we show that ${\rm l.nsp.gldim}_{R}(R)\leq k$ if and only if every
$(n,k)$-weak cotorsion right $R$-module is injective if and only if every
$(n,k)$-weak cotorsion right $R$-module is in $\mathcal{WF}_{k}^{n}(R^{op})$
(see Theorem 4.5), and if every $(n,k)$-weak cotorsion right $R$-module has a
$\mathcal{WF}_{k}^{n}(R^{op})$-envelope with the unique mapping property, then
${\rm l.n.sp.gldim}(R)\leq k+2$ (see Proposition 4.6).

##  2 Preliminaries

In this section, some fundamental concepts are recalled and notations are
stated.

A right $R$-module $M$ is said to be cotorsion [6] if ${\rm
Ext}_{R}^{1}(F,M)=0$ for any flat right $R$-module $F$. For a non-negative
integer $k$, a right $R$-module $M$ is said to be $k$-cotorsion [15] if ${\rm
Ext}_{R}^{1}(F,M)=0$ for any right $R$-module $F\in\mathscr{F}_{k}(R^{op})$. A
left $R$-module $U$ is called super finitely presented [9] if there exists an
exact sequence $\cdots\rightarrow F_{2}\rightarrow F_{1}\rightarrow
F_{0}\rightarrow U\rightarrow 0$, where each $F_{i}$ is finitely generated and
free. A left $R$-module $M$ is called weak injective [10] if ${\rm
Ext}_{R}^{1}(U,M)=0$ for any super finitely presented left $R$-module $U$. A
right $R$-module $M$ is called weak flat [10] if ${\rm Tor}_{1}^{R}(M,U)=0$
for any super finitely presented left $R$-module $U$. For a non-negative
integer $k$, a right $R$-module $M$ is called $k$-weak cotorsion [20] if ${\rm
Ext}_{R}^{1}(F,M)=0$ for any $F\in\mathcal{WF}_{k}(R^{op})$. A left $R$-module
$M$ is said to be FP-injective [21] if ${\rm Ext}_{R}^{1}(N,M)=0$ for any
finitely presented left $R$-module $N$.

######  Definition 2.1 ([1], Definition 2.1).

Let $n$ be a non-negative integer. A left $R$-module $U$ is said to be
$n$-super finitely presented if there exists an exact sequence

$\cdots\to F_{n+1}\to F_{n}\to\cdots\to F_{1}\to F_{0}\to U\to 0$

of projective $R$-modules, where each $F_{i}$ is finitely generated and
projective for any $i\geq n$. If $K_{i}:={\rm Im}(F_{i+1}\to F_{i})$, then for
$i=n-1$, the module $K_{n-1}$ is called special super finitely presented.

Notice that ${\rm Ext}_{R}^{n+1}(U,M)\cong{\rm Ext}_{R}^{1}(K_{n-1},M)$ and
${\rm Tor}_{n+1}^{R}(N,U)\cong{\rm Tor}_{1}^{R}(N,K_{n-1})$, where $U$ is an
$n$-super finitely presented left module with an associated special super
finitely presented module $K_{n-1}$. This fact will be used throughout the
paper.

$0$-Super finitely presented modules are just super finitely presented
modules. Also for any $m\geq n$, every $n$-super finitely presented left
$R$-module is $m$-super finitely presented but not conversely (see Example 2.2
and [1, Examples 2.4 and 2.5]).

The finitely presented dimension of an $R$-module $A$ is defined as ${\rm
f.p.dim}_{R}(A)={\inf}\\{m\mid\text{there exists an exact sequence}\ {\rm
F}_{n+1}\to{\rm F}_{n}\to\cdots\to{\rm F}_{1}\to{\rm F}_{0}\to{\rm A}\to 0$ of
$R$-modules, where each ${\rm F}_{i}$ is projective, and ${\rm F}_{n}$ and
${\rm F}_{n+1}$ are finitely generated$\\}$. We also define the finitely
presented dimension of $R$ (denoted by ${\rm f.p.dim}(R)$) as $\sup\\{{\rm
f.p.dim}_{R}(A)\ \mid\ A\ \text{is a finitely generated}\ R\text{-module}\\}$.

Also, $R$ is called an $(a,b,c)$-ring if ${\rm w.gl.dim}(R)=a$, ${\rm
gl.dim}(R)=b$ and ${\rm f.p.dim}(R)=c$ (see [17]).

######  Example 2.2.

Let $R=R_{1}\oplus R_{2}$, where $R_{1}$ is a ring of polynomials in $4$
indeterminates over a field $k$, and $R_{2}$ is a $(3,3,4)$-ring (see, [1,
Example 2.4] and [17, Proposition 3.10]). Then by [17, Proposition 3.8], $R$
is a coherent $(4,4,4)$-ring. Then, ${\rm f.p.dim}(R)=4$ and so there exists a
finitely generated $R$-module $U$ with ${\rm f.p.dim}_{R}(U)=4$. Hence, there
exists an exact sequence $F_{5}\rightarrow F_{4}\rightarrow F_{3}\rightarrow
F_{2}\rightarrow F_{1}\rightarrow F_{0}\rightarrow U\rightarrow 0,$ where
$F_{4}$ and $F_{5}$ are finitely generated and projective $R$-modules. Also,
$K_{3}:={\rm Im}(F_{4}\rightarrow F_{3})$ is a special super finitely
presented module, since $R$ is coherent. So by Definition 2.1, $U$ is
$4$-super finitely presented. But, every $4$-super finitely presented is not
$3$-super finitely presented otherwise ${\rm f.p.dim}(R)=3$, a contradiction.

######  Definition 2.3 ([1], Definition 2.2).

Let $n$ be a non-negative integer. Then, a left $R$-module $M$ is called
$n$-weak injective if ${\rm Ext}_{R}^{n+1}(U,M)=0$ for every $n$-super
finitely presented left $R$-module $U$. A right $R$-module $N$ is called
$n$-weak flat if ${\rm Tor}_{n+1}^{R}(N,U)=0$ for every $n$-super finitely
presented left $R$-module $U$.

$0$-Weak injective modules are just weak injective modules and $0$-weak flat
modules are just weak flat modules. And for any $m\geq n$, every $n$-weak
injective and every $n$-weak flat module is $m$-weak injective and $m$-weak
flat respectively, but not conversely (see Example 3.3(1)) and [1, Example
2.5(2)].

The left super finitely presented dimension ${\rm l.sp.gldim}(R)$ of $R$ is
defined as ${\rm l.sp.gldim}(R):={\rm sup}\\{{\rm pd}_{R}(M)\mid M\ \text{is a
super finitely presented left}\ R\text{-module}\\}$, whereas the $n$-super
finitely presented dimension of a ring $R$ is defined as follows: ${\rm
l.n.sp.gldim}(R):={\rm sup}\\{{\rm pd}_{R}(K_{n-1})\mid K_{n-1}\ \text{is a
special super finitely presented left}\ R\text{-module}\\}$ (see [1,
Definition 4.9]).

It is easy to see that for any $n\geq 0$, ${\rm l.n.sp.gldim}(R)\leq{\rm
l.sp.gldim}(R)$. Furthermore, Example 3.3(3) shows that we do not have an
equality, but if $n=0$, then clearly ${\rm l.n.sp.gldim}(R)={\rm
l.sp.gldim}(R)$.

In [11, Theorem 3.8], it is proved that ${\rm l.sp.gldim}(R)\leq{\rm
w.gl.dim}(R)$. Then, it is easy to obtain the following result.

######  Proposition 2.4.

For any $n\geq 0$, ${\rm l.n.sp.gldim}(R)\leq{\rm w.gl.dim}(R)$.

Let ${\cal F}$ be a class of $R$-modules and $M$ be an $R$-module. Following
[7], we say that a morphism $f:F\to M$ is an ${\cal F}$-precover of $M$ if
$F\in{\cal F}$ and ${{\rm Hom}}_{R}(F^{{}^{\prime}},F)\to{{\rm
Hom}}_{R}(F^{{}^{\prime}},M)\to 0$ is exact for any $F^{\prime}\in{\cal F}$.
An ${\cal F}$-precover $f:F\to M$ is said to be an ${\cal F}$-cover if every
endomorphism $g:X\to X$ such that $fg=f$ is an isomorphism. Dually, we have
the definitions of an ${\cal F}$-preenvelope and an ${\cal F}$-envelope. The
class ${\cal F}$ is called (pre)covering if each $R$-module has an ${\cal
F}$-(pre)cover. Similarly, if every $R$-module has an ${\cal
F}$-(pre)envelope, then we say that ${\cal F}$ is (pre)enveloping.

To any given class of right $R$-modules $\mathcal{L}$ and class of left
$R$-modules $\mathcal{L^{\prime}}$, we associate its orthogonal classes as
follows:

${\mathcal{L}}^{\bot}={{\rm Ker}Ext}_{R}^{1}(\mathcal{L},-)=\\{C\in
R\text{-}{\rm Mod}\mid{{\rm Ext}}_{R}^{1}(L,C)=0\ \text{for any}\
L\in\mathcal{L}\\},$ ${}^{\bot}{\mathcal{L}}={{\rm
Ker}Ext}_{R}^{1}(-,\mathcal{L})=\\{C\in R\text{-}{\rm Mod}\mid{\rm
Ext}_{R}^{1}(C,L)=0\ \text{for any}\ L\in\mathcal{L}\\},$
${\mathcal{L}}^{\top}={{\rm Ker}{\rm Tor}}^{R}_{1}(\mathcal{L},-)=\\{C\in
R\text{-}{\rm Mod}\mid{\rm Tor}^{R}_{1}(L,C)=0\ \text{for any}\
L\in\mathcal{L}\\},$ ${}^{\top}{\mathcal{L}^{\prime}}={{\rm
Ker}Tor}^{R}_{1}(-,\mathcal{L}^{\prime})=\\{C\in{\rm Mod}\text{-}R\mid{\rm
Tor}^{R}_{1}(C,L)=0\ \text{for any}\ L\in\mathcal{L^{\prime}}\\}.$

A pair $({\cal F},\mathcal{C})$ of classes of $R$-modules is called a
cotorsion theory [7] if ${\cal F}^{\bot}=\mathcal{C}$ and ${\cal
F}={{}^{\bot}\mathcal{C}}$. A cotorsion theory $({\cal F},\mathcal{C})$ is
called hereditary if whenever $0\to F^{{}^{\prime}}\to F\to
F^{{}^{\prime\prime}}\to 0$ is exact with $F,F^{{}^{\prime\prime}}\in{\cal F}$
then $F^{{}^{\prime}}\in{\cal F}$, or equivalently, if $0\to
C^{{}^{\prime}}\to C\to C^{{}^{\prime\prime}}\to 0$ is an exact sequence with
$C,C^{{}^{\prime}}\in\mathcal{C}$, then $C^{{}^{\prime\prime}}\in\mathcal{C}$.
A cotorsion theory $({\cal F},\mathcal{C})$ is called complete [22] if every
$R$-module has a special $\mathcal{C}$-preenvelope; that is, a monic
$\mathcal{C}$-preenvelope with cokernel in ${}^{\bot}\mathcal{C}$ (and a
special ${\cal F}$-precover; that is, an epic ${\cal F}$-precover with kernel
in ${\cal F}^{\bot}$). A cotorsion theory $({\cal F},\mathcal{C})$ is called
perfect if every $R$-module has a $\mathcal{C}$-envelope and an ${\cal
F}$-cover (see [8, 12]).

######  Definition 2.5 ([1], Definition 3.1).

Let $n$ be a non-negative integer. Then, the $n$-weak injective dimension of a
left module $M$ is defined by:

$n$-${\rm wid}_{R}(M):={\inf}\\{k:\ {{\rm Ext}}_{R}^{k+1}(K_{n-1},M)=0\
\text{for every special super finitely presented left module}\ K_{n-1}\\}$,
and the n-weak flat dimension of a right module N is defined by:

$n$-${\rm wfd}_{R}(N):=\inf\\{k:\ {\rm Tor}^{R}_{k+1}(N,K_{n-1})=0\ \text{for
every special super finitely presented left module}\ K_{n-1}\\}$.

The classes of left and right $R$-modules with $n$-weak injective and $n$-weak
flat dimensions at most $k$ will be denoted respectively by
$\mathcal{WI}_{k}^{n}(R)$ and $\mathcal{WF}_{k}^{n}(R^{op})$. In [1, Theorem
4.4 and 4.5], it is proved that $\mathcal{WI}_{k}^{n}(R)$ and
$\mathcal{WF}_{k}^{n}(R^{op})$ are both covering and preenveloping. Here, we
give more properties of these two classes of modules. We start with the
following result.

######  Proposition 2.6.

Let $n$ and $k$ be non-negative integers. Then, the following assertions hold:

  1. (1)

If $M$ is the cokernel of a $\mathcal{WI}_{k}^{n}(R)$-preenvelope $X\to F$ of
a left R-module $X$ with $F$ flat, then
$M\in{}^{\top}\mathcal{WF}_{k}^{n}(R^{op})$.

  2. (2)

If $M$ is the cokernel of a $\mathcal{WF}_{k}^{n}(R^{op})$-preenvelope $A\to
F$ of a right R-module $A$ with $F$ flat, then
$M\in\mathcal{WI}_{k}^{n}(R)^{\top}$.

###### Proof.

(1) Consider the short exact sequence $0\to X\to F\to M\to 0$. If
$N\in\mathcal{WF}_{k}^{n}(R^{op})$, then by [1, Proposition 3.6],
$N^{*}\in\mathcal{WI}_{k}^{n}(R)$. Thus, ${{\rm Hom}}_{R}(F,N^{*})\to{{\rm
Hom}}_{R}(X,N^{*})\to 0$ is exact and so
$(F\otimes_{R}N)^{*}\to(X\otimes_{R}N)^{*}\to 0$ is exact. Hence, $0\to
X\otimes_{R}N\to F\otimes_{R}N$ is exact and from the exactness of the long
exact sequence $0\to{\rm Tor}_{1}^{R}(M,N)\to X\otimes_{R}N\to F\otimes_{R}N$
(because $F$ is flat), it follows that ${\rm Tor}_{1}^{R}(M,N)=0$ and so
$M\in{}^{\top}\mathcal{WF}_{k}^{n}(R^{op})$.

(2) Consider the exact sequence $0\to X\to F\to M\to 0$, where $X={\rm
Im}(A\to F)$. By using similar arguments in (1), we prove that
$M\in\mathcal{WI}_{k}^{n}(R)^{\top}$. ∎

######  Corollary 2.7.

Let $n$ and $k$ be non-negative integers. Then,
${}^{\bot}\mathcal{WF}_{k}^{n}(R^{op})\subseteq\mathcal{WI}_{k}^{n}(R)^{\top}$.

###### Proof.

Consider an exact sequence $0\to X\to F\to N\to 0$, where
$N\in{}^{\bot}\mathcal{WF}_{k}^{n}(R^{op})$ and $F$ is projective. Then,
${{\rm Ext}}_{R}^{1}(N,M)=0$ for any $M\in\mathcal{WF}_{k}^{n}(R^{op})$.
Hence, $X\to F$ is a $\mathcal{WF}_{k}^{n}(R^{op})$-preenvelope of $X$, and so
$N\in\mathcal{WI}_{k}^{n}(R)^{\top}$ by Proposition 2.6(2). ∎

######  Theorem 2.8.

Let $n$ and $k$ be non-negative integers and $U$ be an $n$-super finitely
presented left module. If $\cdots\to F_{n+1}\to F_{n}\to\cdots\to F_{1}\to
F_{0}\to U\to 0$ is a projective resolution such that for any $i\geq n$,
$F_{i}$ is finitely generated and projective and $K_{i}={\rm Im}(F_{i+1}\to
F_{i})$, then the following assertions hold:

  1. (1)

If ${\rm n\text{-}wid}(_{R}R)\leq k$, then
$K_{n-1}\in{{}^{\top}}\mathcal{WF}_{k}^{n}(R^{op})$ if and only if $K_{n}\to
F_{n}$ is a $\mathcal{WI}_{k}^{n}(R)$-preenvelope.

  2. (2)

$K_{n-1}\in\mathcal{WI}_{k}^{n}(R)^{\top}$ if and only if $K_{n}\to F_{n}$ is
a $\mathcal{WF}_{k}^{n}(R^{op})$-preenvelope if and only if
$K_{n-1}\in{{}^{\bot}}\mathcal{WF}_{k}^{n}(R^{op})$.

###### Proof.

(1) Consider the exact sequence $0\to K_{n}\to F_{n}\to K_{n-1}\to 0$ with
$F_{n}$ is finitely generated and projective. It is clear by [1, Proposition
4.8] that $F_{n}\in\mathcal{WI}_{k}^{n}(R)$ (because ${\rm
n\text{-}wid}(_{R}R)\leq k$). We show that $K_{n}\to F_{n}$ is a
$\mathcal{WI}_{k}^{n}(R)$-preenvelope. For any $X\in\mathcal{WI}_{k}^{n}(R)$,
we have $X^{*}\in\mathcal{WF}_{k}^{n}(R^{op})$ by [1, Proposition 3.6(2)]. So
${\rm Tor}^{R}_{1}(K_{n-1},X^{*})=0$. By [18, Lemma 3.55], we obtain the
following commutative diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{n}\otimes_{R}X^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{f_{2}}$$\textstyle{F_{n}\otimes_{R}X^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{1}}$$\textstyle{{\rm
Hom}_{R}(K_{n},X)^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{{\rm
Hom}_{R}(F_{n},X)^{*}.}$

where $f_{1}$ and $f_{2}$ are isomorphisms. We deduce that $\beta$ is a
monomorphism and consequently the sequence ${\rm Hom}_{R}(F_{n},X)\to{\rm
Hom}_{R}(K_{n},X)\to 0$ is exact. The converse follows from Proposition
2.6(1).

(2) The proof of the first equivalence is similar to that of (1) using
Proposition 2.6(2). The second equivalence is clear using Corollary 2.7(2). ∎

##  3 $(n,k)$-Weak cotorsion modules

In this section, we introduce and study $(n,k)$-weak cotorsion modules. We
start with the definition.

######  Definition 3.1.

Let $n$ and $k$ be non-negative integers. Then, a right $R$-module $M$ is
called $(n,k)$-weak cotorsion if ${\rm Ext}_{R}^{1}(N,M)=0$ for every
$N\in\mathcal{WF}_{k}^{n}(R^{op})$.

######  Remark 3.2.

  1. (1)

$(0,k)$-Weak cotorsion right modules are exactly $k$-weak cotorsion modules,
and $(0,0)$-weak cotorsion right modules are exactly weak cotorsion modules.

  2. (2)

Every $(n,k)$-weak cotorsion right module is $(m,k)$-weak cotorsion for any
$n\geq m$, but not conversely. Indeed, if every $(m,k)$-weak cotorsion right
module $M$ is $(n,k)$-weak cotorsion, then either $M$ is injective or for each
$N\in\mathcal{WF}_{k}^{n}(R^{op})$, $N$ is projective or
$N\in\mathcal{WF}_{k}^{n-1}(R^{op})$, and this is impossible (see Example
3.3(2)).

  3. (3)

The study of the class of $(n,k)$-weak cotorsion right modules involve the
classes of injective, $(n,k)$-weak cotorsion, $k$-weak cotorsion,
$k$-cotorsion and cotorsion right $R$-modules denoted $\mathcal{I}$,
$\mathcal{WC}_{k}^{n}(R^{op})$, $\mathcal{WC}_{k}(R^{op})$,
$\mathcal{C}_{k}(R^{op})$ and $\mathcal{C}(R^{op})$ respectively. In fact,

$\mathcal{I}\subseteq\mathcal{WC}_{k}^{n}(R^{op})\subseteq\mathcal{WC}_{k}(R^{op})\subseteq\mathcal{C}_{k}(R^{op})\subseteq\mathcal{C}(R^{op}).$

But not every $k$-cotorsion right module is $(n,k)$-weak cotorsion (see
Example 3.3(3)).

######  Example 3.3.

Let $R=k[x]\times S$, where $k[x]$ is a ring of polynomials in one
indeterminate over a field $k$ and $S$ is a non-Noetherian hereditary von
Neumann regular ring. For example, $S$ is a ring of functions of $X$ into $k$
continuous with respect to the discrete topology on $k$, where $k$ is a field
and $X$ is a totally disconnected compact Hausdorff space whose associated
Boolean ring is hereditary (see examples of [4]). Then, by [17, Proposition
3.10], $R$ is a coherent $(1,1,2)$-ring. Then, we have:

  1. (1)

Every $R$-module is $1$-weak flat, since ${\rm gl.dim}(R)=1$. But not every
$R$-module is $0$-weak flat, for otherwise each super finitely presented
$R$-module would be projective. On the other hand, every finitely presented is
super finitely presented (because $R$ is coherent). Thus every finitely
presented is flat and by [25, Theorem 3.9], each $R$-module is flat and so
${\rm w.gl.dim}(R)=0$, a contradiction.

  2. (2)

Notice that ${\rm l.n.sp.gldim}(R)=0$ for any $n\geq 1$, since ${\rm
pd}_{R}(U)\leq 1$ for any $n$-super finitely presented $R$-module $U$. So by
Theorem 4.5, every $(n,0)$-weak cotorsion right $R$-module is injective for
any $n\geq 1$. But not every $(0,0)$-weak cotorsion right $R$-module is
injective, for otherwise every $R$-module would be in
$\mathcal{WF}_{0}^{0}(R^{op})$ and so each $0$-super finitely presented would
be flat. Similarly to (1), ${\rm w.gl.dim}(R)=0$, a contradiction.

  3. (3)

Also, ${\rm l.sp.gldim}(R)=1$, since $R$ is coherent and by [11, Theorem 3.8],
${\rm l.sp.gldim}(R)={\rm w.gl.dim}_{R}(R)=1$. So ${\rm
l.n.sp.gldim}(R)\neq{\rm l.sp.gldim}(R)$ and it follows by Proposition 3.4
that not every $k$-cotorsion is $(n,k)$-weak cotorsion.

Example 3.3(3) gives a context when ${\rm l.nsp.gldim}(R)\leq{\rm
l.sp.gldim}(R)$.

Now we show, for non-negative integers $k$ and $n$, when every $k$-cotorsion
module is $(n,k)$-weak cotorsion.

######  Proposition 3.4.

Let $n$ be a non-negative integer. Then, the following conditions are
equivalent:

  1. (1)

For every non-negative integer $k$, every $k$-cotorsion right module is
$(n,k)$-weak cotorsion.

  2. (2)

For every non-negative integer $k$,
$\mathcal{WF}_{k}^{n}(R^{op})=\mathscr{F}_{k}(R^{op})$.

  3. (3)

${\rm l.n.sp.gldim}(R)={\rm w.gl.dim}(R)$.

###### Proof.

The implication $(1)\Rightarrow(2)$ follows from the fact that
$(\mathcal{WF}_{k}^{n}(R^{op})),\mathcal{WC}_{k}^{n}(R^{op}))$ and
$(\mathscr{F}_{k}(R^{op}),\mathcal{C}_{k}(R^{op}))$ are cotorsion theories.
And, the implications $(2)\Leftrightarrow(3)\Rightarrow(1)$ are clear. ∎

The following result can be easily obtained.

######  Proposition 3.5.

Let $n$ and $k$ be non-negative integers. Then, the following assertions hold:

  1. (1)

If $\\{M_{i}\\}_{i\in I}$ is a family of right $R$-modules, then $\prod_{i\in
I}M_{i}$ is $(n,k)$-weak cotorsion if and only if each $M_{i}$ is $(n,k)$-weak
cotorsion.

  2. (2)

The class $\mathcal{WC}_{k}^{n}(R^{op})$ is closed under extensions and direct
summands.

Now we state one of the main results of this section which characterizes when
a module $M$ is $(n,k)$-weak cotorsion. We start with the following
proposition.

######  Proposition 3.6.

Let $n$ and $k$ be non-negative integers and let $M$ be a right $R$-module.
Then, the following conditions are equivalent:

  1. (1)

$M$ is $(n,k)$-weak cotorsion.

  2. (2)

$M$ is injective with respect to every exact sequence $0\to K\to N\to D\to 0$
of right $R$-modules, where $D\in\mathcal{WF}_{k}^{n}(R^{op})$.

Moreover if ${\rm n\text{-}wid}(_{R}R)\leq k$, then the above conditions are
also equivalent to:

  1. (3)

For every exact sequence $0\to M\to E\to D\to 0$ of right $R$-modules, where
$E$ is injective, $E\to D$ is a $\mathcal{WF}_{k}^{n}(R^{op})$-precover of
$D$.

  2. (4)

$M$ is the kernel of a $\mathcal{WF}_{k}^{n}(R^{op})$-precover
$E\stackrel{{\scriptstyle\displaystyle\phi}}{{\to}}D$ where $E$ is injective.

###### Proof.

$(1)\Rightarrow(2)$ Clear.

$(2)\Rightarrow(1)$ Let $D\in\mathcal{WF}_{k}^{n}(R^{op})$ and consider an
exact sequence $0\rightarrow K\rightarrow P\rightarrow D\rightarrow 0$ where
$P$ projective. Then, from the long exact sequence ${{\rm
Hom}}_{R}(P,M)\rightarrow{{\rm Hom}}_{R}(K,M)\rightarrow{\rm
Ext}_{R}^{1}(D,M)\rightarrow{\rm Ext}_{R}^{1}(P,M)$ we see that ${\rm
Ext}_{R}^{1}(D,M)=0$ and hence $M$ is $(n,k)$-weak cotorsion.

$(1)\Rightarrow(3)$ For every exact sequence $0\rightarrow M\rightarrow
E\stackrel{{\scriptstyle\displaystyle\phi}}{{\rightarrow}}D\rightarrow 0$
where $E$ is injective, we have by [1, Proposition 4.8] that
$E\in\mathcal{WF}_{k}^{n}(R^{op})$. So for any $N$ in
$\mathcal{WF}_{k}^{n}(R^{op})$, we have

$0\rightarrow{{\rm Hom}}_{R}(N,M)\rightarrow{{\rm
Hom}}_{R}(N,E)\rightarrow{{\rm Hom}}_{R}(N,D)\rightarrow{\rm
Ext}_{R}^{1}(N,M)=0.$

It follows that $\phi$ is a $\mathcal{WF}_{k}^{n}(R^{op})$-precover of $D$.

$(3)\Rightarrow(4)$ Trivial.

$(4)\Rightarrow(1)$ By hypothesis, there exists an exact sequence
$0\rightarrow M\rightarrow
E\stackrel{{\scriptstyle\displaystyle\phi}}{{\rightarrow}}D\rightarrow 0$,
where $\phi$ is a $\mathcal{WF}_{k}^{n}(R^{op})$-precover of $D$. Therefore
for each $L$ in $\mathcal{WF}_{k}^{n}(R^{op})$, we have:

$0\rightarrow{{\rm Hom}}_{R}(L,M)\rightarrow{{\rm
Hom}}_{R}(L,E)\rightarrow{{\rm Hom}}_{R}(L,D)\rightarrow{\rm
Ext}_{R}^{1}(L,M)\rightarrow{\rm Ext}_{R}^{1}(L,E),$

We have ${{\rm Hom}}_{R}(L,E)\rightarrow{{\rm Hom}}_{R}(L,D)$ is epic and
${\rm Ext}_{R}^{1}(L,E)=0$ so ${\rm Ext}_{R}^{1}(L,M)=0$ and consequently $M$
is $(n,k)$-weak cotorsion. ∎

Recall that an $R$-module $N$ is said to be reduced [7] if $N$ has no non zero
injective submodules. In the next proposition, we characterize reduced
$(n,k)$-weak cotorsion modules when ${\rm n\text{-}wid}(_{R}R)\leq k$.

######  Proposition 3.7.

Let $n$ and $k$ be non-negative integers and assume that ${\rm
n\text{-}wid}(_{R}R)\leq k$. Then for a right $R$-module $M$, $M$ is a reduced
$(n,k)$-weak cotorsion if and only if $M$ is the kernel of a
$\mathcal{WF}_{k}^{n}(R^{op})$-cover
$N\stackrel{{\scriptstyle\displaystyle\phi}}{{\rightarrow}}D$ where $N$ is an
injective module.

###### Proof.

Suppose that $M$ is a reduced $(n,k)$-weak cotorsion right module and consider
the exact sequence $0\rightarrow M\rightarrow
E(M)\stackrel{{\scriptstyle\displaystyle\phi}}{{\rightarrow}}\frac{E(M)}{M}\rightarrow
0$. Then, by Proposition 3.6 $\phi$ is a
$\mathcal{WF}_{k}^{n}(R^{op})$-precover. Since $M$ is reduced, $E(M)$ has no
non zero direct summand $L$ contained in $M$. Also by [1, Theorem 4.5],
$\frac{E(M)}{M}$ has a $\mathcal{WF}_{k}^{n}(R^{op})$-cover. Then, [23,
Corollary 1.2.8] implies that $\phi$ is a
$\mathcal{WF}_{k}^{n}(R^{op})$-cover.

Conversely, suppose that $M$ is the kernel of a
$\mathcal{WF}_{k}^{n}(R^{op})$-cover
$N\stackrel{{\scriptstyle\displaystyle\phi}}{{\rightarrow}}D$ where $N$ is an
injective $R$-module. Then by Proposition 3.6, $M$ is $(n,k)$-weak cotorsion.
Let $N_{1}$ be an injective submodule of $M$. If $N=N_{1}\oplus N_{2}$,
$f:N\rightarrow N_{2}$ is the projection map and $i:N_{2}\rightarrow N$ is the
inclusion map, then $\phi(N_{1})=0$ and $\phi(if)=\phi$, and hence $if$ is an
isomorphism. So $i$ is an epimorphism and consequently $N=N_{2}$ and
$N_{1}=0$. Therefore, $M$ is reduced. ∎

Now we are in a position to show the following theorem.

######  Theorem 3.8.

Let $n$ and $k$ be non-negative integers and assume that ${\rm
n\text{-}wid}(_{R}R)\leq k$. Then for a right $R$-module $M$, $M$ is
$(n,k)$-weak cotorsion if and only if $M$ is a direct sum of an injective
right $R$-module and a reduced $(n,k)$-weak cotorsion right $R$-module.

###### Proof.

Consider the exact sequence $0\rightarrow M\rightarrow
E(M)\stackrel{{\scriptstyle\displaystyle\phi}}{{\rightarrow}}\frac{E(M)}{M}\rightarrow
0$, where $M$ is an $(n,k)$-weak cotorsion right module. By Proposition 3.7,
$\phi$ is a $\mathcal{WF}_{k}^{n}(R^{op})$-cover of $\frac{E(M)}{M}$. Also by
[1, Theorem 4.5], $\frac{E(M)}{M}$ has a $\mathcal{WF}_{k}^{n}(R^{op})$-cover
$N\rightarrow\frac{E(M)}{M}$. Then, there exists the commutative diagram with
exact rows:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{1}}$$\scriptstyle{h_{1}}$$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{2}}$$\textstyle{\frac{E(M)}{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{2}}$$\scriptstyle{f_{1}}$$\textstyle{E(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{2}}$$\textstyle{\frac{E(M)}{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{3}}$$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\frac{E(M)}{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

We have $E(M)={\rm ker}f_{2}\oplus{\rm im}h_{2}$, since $f_{2}h_{2}$ is an
isomorphism. Also ${\rm im}h_{2}\cong N$ so $N$ and ${\rm ker}f_{2}$ are
injective. Thus by Proposition 3.7, $N_{1}$ is reduced and $(n,k)$-weak
cotorsion. On the other hand, by the Five Lemma, $f_{1}h_{1}$ is an
isomorphism. Hence, $M={\rm ker}f_{1}\oplus{\rm im}h_{1}$ where ${\rm
im}h_{1}\cong N_{1}$. Consider the following commutative diagram:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
ker}f_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
ker}f_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{2}}$$\scriptstyle{f_{1}}$$\textstyle{E(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{2}}$$\textstyle{\frac{E(M)}{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{3}}$$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\frac{E(M)}{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$

Consequently ${\rm ker}f_{1}\cong{\rm ker}f_{2}$ and so $M=N_{1}\oplus{\rm
ker}f_{1}$, where ${\rm ker}f_{1}$ is injective. The converse follows from
Proposition 3.5(1). ∎

We end this section with a result that shows when every right $R$-module is
$(n,k)$-weak cotorsion. To this end, we prove first the following proposition.

######  Proposition 3.9.

The following assertions hold:

  1. (1)

If a right $R$-module $M$ is $(n,k)$-weak cotorsion, then ${\rm
Ext}_{R}^{j\geq m+1}(N,M)=0$ for every $N\in\mathcal{WF}_{m+k}^{n}(R^{op})$.

  2. (2)

The $m$th cosyzygy of every $(n,k)$-weak cotorsion right $R$-module is
$(n,m+k)$-weak cotorsion.

###### Proof.

(1) Let $N\in\mathcal{WF}_{m+k}^{n}(R^{op})$. Then, there is an exact
sequence:

$0\to K_{m}\to P_{m-1}\to\cdots\to P_{1}\to P_{0}\to N\to 0,$

where each $P_{i}$ is projective and $K_{m}\in\mathcal{WF}_{k}^{n}(R^{op})$.
Hence, ${\rm Ext}_{R}^{m+1}(N,M)\cong{\rm Ext}_{R}^{1}(K_{m},M)=0$ (because
$M$ is $(n,k)$-weak cotorsion).

(2) Let $M$ be any $(n,k)$-weak cotorsion right $R$-module and $D^{m}$ the
$m$th cosyzygy of $M$. Then by (1), ${\rm Ext}_{R}^{m+1}(N,M)=0$ for every
$N\in\mathcal{WF}_{m+k}^{n}(R^{op})$. Thus, $0={\rm
Ext}_{R}^{m+1}(N,M)\cong{\rm Ext}_{R}^{1}(N,D^{m})$ and consequently $D^{m}$
is $(n,m+k)$-weak cotorsion. ∎

######  Theorem 3.10.

Let $n$ and $k$ be non-negative integers. Then, the following conditions are
equivalent:

  1. (1)

Every right $R$-module is $(n,k)$-weak cotorsion.

  2. (2)

Every right $R$-module in $\mathcal{WF}_{k}^{n}(R^{op})$ is projective.

  3. (3)

Every right $R$-module in $\mathcal{WF}_{k}^{n}(R^{op})$ is $(n,k)$-weak
cotorsion.

  4. (4)

For any integer $m$, ${\rm Ext}^{i\geq m+1}_{R}(M,N)=0$ for all right
$R$-modules $N$ and all $R$-module $M$ in $\mathcal{WF}_{m+k}^{n}(R^{op})$.

  5. (5)

Every right $R$-module $M$ has a
$\mathcal{WF}_{k}^{n}(R^{op})^{\bot}$-envelope with the unique mapping
property.

  6. (6)

Every projective right $R$-module is $(n,k)$-weak cotorsion.

  7. (7)

$R_{R}$ is $(n,k)$-weak cotorsion and every right $R$-module has a
$\mathcal{WF}_{k}^{n}(R^{op})^{\bot}$-precover.

###### Proof.

$(1)\Leftrightarrow(2)$ Clear because
$(\mathcal{WF}_{k}^{n}(R^{op})),\mathcal{WC}_{k}^{n}(R^{op}))$ and $(Proj,{\rm
Mod}\text{-}R)$ are cotorsion theories.

$(1)\Rightarrow(4)$ Follows from Proposition 3.9(1).

$(4)\Rightarrow(3)$ Take $m=0$.

$(3)\Rightarrow(1)$ Let $M$ be a right $R$-module. Then by [1, Proposition
4.13], there exists an exact sequence $0\rightarrow L\rightarrow
N\stackrel{{\scriptstyle\displaystyle\phi}}{{\rightarrow}}M\rightarrow 0$,
where $\phi$ is $\mathcal{WF}_{k}^{n}(R^{op})$-cover of $M$. Also by [23,
Lemma 2.1.1], $L$ is in
$\mathcal{WF}_{k}^{n}(R^{op})^{\bot}=\mathcal{WC}_{k}^{n}(R^{op})$. Since
$(\mathcal{WF}_{k}^{n}(R^{op}),\mathcal{WF}_{k}^{n}(R^{op})^{\bot})$ is a
hereditary perfect cotorsion pair, we deduce that every right $R$-module $M$
is $(n,k)$-weak cotorsion.

$(1)\Rightarrow(5)$ Clear.

$(5)\Rightarrow(3)$ Let $N\in\mathcal{WF}_{k}^{n}(R^{op})$. Then, the
following commutative diagram exists:

|
---|---
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{0}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\scriptstyle{\gamma\beta}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\textstyle{0,}$$\textstyle{X^{{}^{\prime}}}$

where $\alpha$ and $\gamma$ are
$\mathcal{WF}_{k}^{n}(R^{op})^{\bot}$-envelope. Since
$\gamma\beta\alpha=0=0\alpha$, it follows by assumption that $\gamma\beta=0$.
Consequently $D={\rm im}(\beta)\subseteq{\rm ker}(\gamma)=0$ and so $D=0$.
Hence, $N$ is $(n,k)$-weak cotorsion.

$(1)\Rightarrow(7)$ Clear.

$(7)\Rightarrow(6)$ By [13, Proposition 1],
$\mathcal{WF}_{k}^{n}(R^{op})^{\bot}$ is closed under direct sums. Since $R$
is $(n,k)$-weak cotorsion, it follows that every free right $R$-module is
$(n,k)$-weak cotorsion. Then by Proposition 3.5(2), we deduce that each
projective right $R$-module is $(n,k)$-weak cotorsion.

$(6)\Rightarrow(1)$ Let $M$ be a right $R$-module. Then by [1, Proposition
4.13], there exists an exact sequence $0\rightarrow L\rightarrow
N\stackrel{{\scriptstyle\displaystyle\alpha}}{{\rightarrow}}M\rightarrow 0$,
where $\alpha$ is $\mathcal{WF}_{k}^{n}(R^{op})$-cover of $M$ and
$L\in\mathcal{WF}_{k}^{n}(R^{op})^{\bot}$ by [23, Lemma 2.1.1]. By hypothesis,
every projective right $R$-module is $(n,k)$-weak cotorsion, and then by
Remark 3.2[(1), (2)], every projective right $R$-module is cotorsion. Hence by
[3, Corollary 10], $R$ is perfect. So by [19, Theorem Bass], any flat right
$R$-module $Y$ is projective and from the assumption, $Y$ is $(n,k)$-weak
cotorsion. Since every flat module is $n$-weak flat and $N$ is in
$\mathcal{WF}_{k}^{n}(R^{op})$, we deduce that for any flat resolution of $N$,
there is an exact sequence $0\rightarrow F_{k}\rightarrow
F_{k-1}\rightarrow\cdots\rightarrow F_{1}\rightarrow F_{0}\rightarrow
N\rightarrow 0,$ where each $F_{i}$ is $(n,k)$-weak cotorsion. If $X_{i}={\rm
im}(F_{i}\rightarrow F_{i-1})$ for any $1\leq i\leq k$, then by [1,
Proposition 4.13], every $X_{i}$ is $(n,k)$-weak cotorsion, and so the exact
sequence $0\rightarrow X_{1}\rightarrow F_{0}\rightarrow N\rightarrow 0$
implies that $N$ is $(n,k)$-weak cotorsion (because $X_{1}$ and $F_{0}$ are
$(n,k)$-weak cotorsion). Similarly, $M$ is $(n,k)$-weak cotorsion with respect
to the exact sequence $0\rightarrow L\rightarrow
N\stackrel{{\scriptstyle\displaystyle\alpha}}{{\rightarrow}}M\rightarrow 0$. ∎

##  4 Applications

In this section, we characterize rings with finite $n$-super finitely
presented dimension in terms of $n$-weak injective and $n$-weak flat modules
and $(n,k)$-weak cotorsion modules.
First, we start with the following lemmas.

######  Lemma 4.1.

Let $n$ and $k$ be non-negative integers. Then, the following conditions are
equivalent for a left $R$-module $N$:

  1. (1)

$N\in\mathcal{WI}_{k}^{n}(R)^{\top}$.

  2. (2)

$N^{*}\in\mathcal{WI}_{k}^{n}(R)^{\bot}$.

  3. (3)

$N\in^{\bot}\rm\mathscr{X},$ where $\rm\mathscr{X}=\\{A^{*}\ \mid\ A\ is\ in\
\mathcal{WI}_{k}^{n}(R)\\}$.

  4. (4)

The functor $-\otimes_{R}N$ preserves the exactness of every exact sequence
$0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$ where
$C\in\mathcal{WI}_{k}^{n}(R)$.

###### Proof.

By [18, Theorem 11.55], for any right $R$-module $D$, ${\rm
Ext}_{R}^{1}(N,D^{*})\cong{\rm Tor}_{1}^{R}(D,N)^{*}\cong{\rm
Ext}_{R}^{1}(D,N^{*})$. So $(1)\Leftrightarrow(2)\Leftrightarrow(3)$ follows.
Also, $(1)\Leftrightarrow(4)$ is clear. ∎

######  Lemma 4.2.

Assume that ${\rm n\text{-}wid}(_{R}R)\leq k$ where $k\geq 1$. Then,

  1. (1)

If $N\in\mathcal{WI}_{k-1}^{n}(R)^{\bot}$, then there is an exact sequence
$0\rightarrow X\rightarrow E\rightarrow N\rightarrow 0$ where $E$ is injective
and $X\in\mathcal{WI}_{k}^{n}(R)^{\bot}$.

  2. (2)

If $N\in\mathcal{WI}_{k-1}^{n}(R)^{\top}$, then there is an exact sequence
$0\rightarrow N\rightarrow F\rightarrow D\rightarrow 0$ where $F$ is flat and
$D\in\mathcal{WI}_{k}^{n}(R)^{\top}$.

###### Proof.

(1) Consider an exact sequence $0\rightarrow Y\rightarrow F\rightarrow
N\rightarrow 0$ of left $R$-modules, where $F$ is projective. Then, we have
the following pushout diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E(F)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0}$

where $\alpha$ is an injective envelope of $F$. Let $U$ be an $n$-super
finitely presented left $R$-module with associated special super finitely
presented module $K_{n-1}$. Then by assumption and from [1, Proposition 3.2],
${\rm Ext}_{R}^{k+1}(K_{n-1},R)=0$ so ${\rm Ext}_{R}^{k+1}(K_{n-1},F)=0$ and
hence ${\rm n\text{-}wid}_{R}(F)\leq k$. Also by [1, Proposition 3.2], it
follows that ${\rm n\text{-}wid}_{R}(B)\leq k-1$, and so
$B\in\mathcal{WI}_{k-1}^{n}(R)$. Hence, ${\rm Ext}_{R}^{1}(B,N)=0$, and
consequently the sequence $0\rightarrow N\rightarrow L\rightarrow B\rightarrow
0$ splits and then $N$ is a quotient of $E(F)$.

By [10, Theorem 3.1], there is a weak injective cover of left $R$-module $N$.
If $\gamma:E\rightarrow N$ is a weak injective cover of $N$, then $\gamma$ is
epic. Now consider the exact sequence $0\rightarrow X\rightarrow E\rightarrow
N\rightarrow 0$. By [23, Lemma 2.1.1], $X\in\mathcal{WI}_{0}^{0}(R)^{\bot}$.
We prove that $X\in\mathcal{WI}_{k}^{n}(R)^{\bot}$. For let $L_{1}$ be in
$\mathcal{WI}_{k}^{n}(R)$ and consider the exact sequence $0\rightarrow
L_{1}\rightarrow E(L_{1})\rightarrow L_{2}\rightarrow 0$, where
$L_{2}\in\mathcal{WI}_{k-1}^{n}(R)$ by [1, Proposition 3.2]. Then we get the
induced exact sequence

${\rm Exr}^{1}_{R}(L_{2},N)\to{\rm Exr}^{2}_{R}(L_{2},X)\to{\rm
Exr}^{2}_{R}(L_{2},E)=0.$

By hypothesis, we have ${\rm Exr}^{1}_{R}(L_{2},N)=0$ and so ${\rm
Exr}^{2}_{R}(L_{2},X)=0$. Also, we obtain the induced exact sequence

${\rm Exr}^{1}_{R}(E(L_{1}),X)\to{\rm Exr}^{1}_{R}(L_{1},X)\to{\rm
Exr}^{2}_{R}(L_{2},X).$

We have ${\rm Exr}^{2}_{R}(L_{2},X)=0$ and since $X$ is in
$\mathcal{WI}_{0}^{0}(R)^{\bot}$ and $E(L_{1})\in\mathcal{WI}_{0}^{0}(R)$, it
follows that ${\rm Exr}^{1}_{R}(E(L_{1}),X)=0$. Hence, ${\rm
Exr}^{1}_{R}(L_{1},X)=0$ and consequently
$X\in\mathcal{WI}_{k}^{n}(R)^{\bot}$.

(2) Consider an exact sequence $0\rightarrow K\rightarrow F\rightarrow
N^{*}\rightarrow 0$ of left $R$-modules where $F$ is projective. Then, we have
the following pushout diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{N^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E(F)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0}$

where $\alpha$ is an injective envelope of $F$. By Lemma 4.1,
$N^{*}\in\mathcal{WI}_{k}^{n}(R)^{\bot}$. Then similar to the proof of (1), we
see ${\rm Ext}_{R}^{1}(Z,N^{*})=0$, and hence the sequence $0\rightarrow
N^{*}\rightarrow L\rightarrow Z\rightarrow 0$ splits. Thus, $N^{*}$ is a
quotient of $E(F)$ and then we have the exact sequence $0\rightarrow
N^{**}\rightarrow(E(F))^{*}$ exists with $(E(F))^{*}$ is flat. By [23,
Proposition 2.3.5], $N$ is pure in $N^{**}$, and hence $N$ embeds in flat
$R$-module.

Let $\beta:N\rightarrow F$ be a flat preenvelope of $N$. Then, $\beta$ is
monic, and so we have the exact sequence $0\rightarrow N\rightarrow
F\rightarrow D\rightarrow 0$. By [23, Lemma 2.1.2],
$D\in{{}^{\bot}\mathscr{F}_{0}(R^{op})}\subseteq{{}^{\bot}\mathcal{WF}_{0}^{0}(R^{op})}$,
and then by Corollary 2.7, $D$ is in $\mathcal{WI}_{0}^{0}(R)^{\top}$. We show
that $D\in\mathcal{WI}_{k}^{n}(R)^{\top}$. Now, let $K_{1}$ be in
$\mathcal{WI}_{k}^{n}(R)$. Then, there exists an exact sequence $0\rightarrow
K_{1}\rightarrow E(K_{1})\rightarrow K_{2}\rightarrow 0$, where
$K_{2}\in\mathcal{WI}_{k-1}^{n}(R)$ by [1, Proposition 3.2]. Then we get the
exact sequence

$0={\rm Tor}_{2}^{R}(K_{2},F)\to{\rm Tor}_{2}^{R}(K_{2},D)\to{\rm
Tor}_{1}^{R}(K_{2},N).$

We have ${\rm Tor}_{1}^{R}(K_{2},N)=0$ and so ${\rm Tor}_{2}^{R}(K_{2},D)=0$.
Also, we have the following exact sequence

$0={\rm Tor}_{2}^{R}(K_{2},D)\to{\rm Tor}_{1}^{R}(K_{1},D)\to{\rm
Tor}_{1}^{R}(E(K_{1}),D).$

Since $D$ is in $\mathcal{WI}_{0}^{0}(R)^{\top}$ and
$E(K_{1})\in\mathcal{WI}_{0}^{0}(R)$, we deduce that ${\rm
Tor}_{1}^{R}(E(K_{1}),D)=0$. Hence, ${\rm Tor}_{1}^{R}(K_{1},D)=0$ and then
$D\in\mathcal{WI}_{k}^{n}(R)^{\top}$. ∎

######  Lemma 4.3.

Assume that ${\rm l.nsp.gldim}(R)<\infty$. Then, ${\rm l.nsp.gldim}(R)$=${\rm
n\text{-}wid}(_{R}R)$.

###### Proof.

By [1, Proposition 3.5 ], ${\rm l.nsp.gldim}(R)=\sup\\{n$-${\rm wid}_{R}(M)\
\mid\ {\rm M\ is\ a\ left}\ R$-$module\\}={\sup}\\{n$-${\rm wfd}_{R}(M)\ \mid\
{\rm M\ is\ a\ right}\ R$-$module\\}$. So, we have ${\rm
n\text{-}wid}(_{R}R)\leq{\rm l.nsp.gldim}(R)$. Now, let ${\rm
l.nsp.gldim}(R)=k.$ We show that ${\rm n\text{-}wid}(_{R}R)\geq k.$ For let
$U$ be an $n$-super finitely presented left $R$-module with associated special
super finitely presented module $K_{n-1}$. So there exists left $R$-module $M$
such that ${\rm Ext}_{R}^{k}(K_{n-1},M)\neq 0$. Consider an exact sequence
$0\rightarrow L\rightarrow F\rightarrow M\rightarrow 0$ where $F$ is free.
Then we have the following exact sequence:

${\rm Ext}_{R}^{k}(K_{n-1},F)\rightarrow{\rm
Ext}_{R}^{k}(K_{n-1},M)\rightarrow{\rm Ext}_{R}^{k+1}(K_{n-1},L).$

We have ${\rm Ext}_{R}^{k+1}(K_{n-1},L)$ (because ${\rm l.nsp.gldim}(R)=k$).
Therefore ${\rm Ext}_{R}^{k}(K_{n-1},R)\neq 0$, otherwise ${\rm
Ext}_{R}^{k}(K_{n-1},F)=0$ and so from the exact sequence above, ${\rm
Ext}_{R}^{k}(K_{n-1},M)=0$ which is a contradition. Consequently ${\rm
n\text{-}wid}(_{R}R)\geq k$. ∎

The following theorem extends the results of Mao and Ding [14, Theorem 6.4 ]
and of Selvaraj and Prabakaran [20, Theorem 6].

######  Theorem 4.4.

Let $n$ be a non-negative integer and assume that ${\rm
n\text{-}wid}(_{R}R)\leq k$ where $k\geq 1$. Then, the following conditions
are equivalent:

  1. (1)

${\rm l.n.sp.gldim}(R)<\infty$.

  2. (2)

${\rm l.n.sp.gldim}(R)\leq k$.

  3. (3)

Every left $R$-module in ${}^{\bot}\mathcal{WI}_{k}^{n}(R)$ belongs to
$\mathcal{WI}_{k}^{n}(R)$.

  4. (4)

Every left $R$-module in $\mathcal{WI}_{k-1}^{n}(R)^{\bot}$ is injective.

  5. (5)

Every left $R$-module in $\mathcal{WI}_{k-1}^{n}(R)^{\bot}$ is $n$-weak
injective.

  6. (6)

Every left $R$-module in $\mathcal{WI}_{k}^{n}(R)^{\bot}$ is injective.

  7. (7)

Every left $R$-module in $\mathcal{WI}_{k}^{n}(R)^{\bot}$ is $n$-weak
injective.

  8. (8)

Every left $R$-module in $\mathcal{WI}_{k}^{n}(R)^{\bot}$ belongs to
$\mathcal{WI}_{k}^{n}(R)$.

  9. (9)

Every left $R$-module (with $M\in{{}^{\bot}\mathcal{WI}_{k-1}^{n}(R)}$) has a
monic $\mathcal{WI}_{k-1}^{n}(R)$-cover.

If $R$ is a left coherent ring, then the above conditions are also equivalent
to:

  1. (10)

Every right $R$-module in $\mathcal{WI}_{k-1}^{n}(R)^{\top}$ is flat.

  2. (11)

Every right $R$-module in $\mathcal{WI}_{k-1}^{n}(R)^{\top}$ is $n$-weak flat.

  3. (12)

Every right $R$-module in $\mathcal{WI}_{k}^{n}(R)^{\top}$ is flat.

  4. (13)

Every right $R$-module in $\mathcal{WI}_{k}^{n}(R)^{\top}$ is $n$-weak flat.

###### Proof.

$(2)\Rightarrow(1)$, $(4)\Rightarrow(6)\Rightarrow(7)$,
$(4)\Rightarrow(5)\Rightarrow(7)$, $(10)\Rightarrow(11)\Rightarrow(13)$ and
$(10)\Rightarrow(12)\Rightarrow(13)$ are clear.

$(1)\Rightarrow(2)$ Clear by Lemma 4.3.

$(2)\Leftrightarrow(3)\Leftrightarrow(9)$ and $(2)\Rightarrow(8)$ Follow from
[1, Proposition 4.10].

$(8)\Rightarrow(2)$ Let $M$ be a left $R$-module. Since ${\rm
n\text{-}wid}(_{R}R)\leq k$, it follows that $R\in\mathcal{WI}_{k}^{n}(R)$.
Then by [1, Proposition 4.8], there is a short exact sequence $0\rightarrow
K\rightarrow F\stackrel{{\scriptstyle\displaystyle
h}}{{\rightarrow}}M\rightarrow 0$, where $h$ is an epic
$\mathcal{WI}_{k}^{n}(R)$-cover of $M$. Hence by [23, Lemma 2.1.1] and [1,
Proposition 4.12], we deduce that $K\in\mathcal{WI}_{k}^{n}(R)^{\bot}$. By
assumption, $K\in\mathcal{WI}_{k}^{n}(R)$, and so [1, Corollary 3.4 ] implies
that $M\in\mathcal{WI}_{k}^{n}(R)$. Consequently, by [1, Proposition 4.10],
${\rm l.n.sp.gldim}(R)\leq k$.

$(6)\Rightarrow(4)$ Clear by Lemma 4.2(1).

$(7)\Rightarrow(5)$ Clear by Lemma 4.2(1) and [1, Corollary 3.4(1)].

$(2)\Leftrightarrow(6)$ Follows from [1, Propositions 4.10 and 4.12 (1)].

$(5)\Rightarrow(4)$ Let $M\in\mathcal{WI}_{k-1}^{n}(R)^{\bot}$. By assumption,
$M$ is also $n$-weak injective. Now consider an exact sequence $0\rightarrow
M\rightarrow E\rightarrow D\rightarrow 0$, where $E$ is injective and so also
$n$-weak injective. Then by [1, Corollary 3.4 ], $D$ is $n$-weak injective. By
[1, Theorem 2.13(1)], ${\rm Ext}_{R}^{k}(K_{n-1},D)=0$ for any special super
finitely presented $K_{n-1}$, and hence by [1, Proposition 3.2],
$D\in\mathcal{WI}_{k-1}^{n}(R)$. Then, ${\rm Ext}_{R}^{1}(D,M)=0$. So the
above exact sequence splits and consequently $M$ is injective.

$(7)\Rightarrow(6)$ Similar to the proof of $(5)\Rightarrow(4)$.

$(12)\Rightarrow(10)$ Trivial by Lemma 4.2(2).

$(13)\Rightarrow(11)$ Trivial by Lemma 4.2(2) and [1, Corollary 3.4(2)].

$(13)\Rightarrow(12)$ Let $M$ be in $\mathcal{WI}_{k}^{n}(R)^{\top}$. Then by
(13), $M$ is $n$-weak flat. By Lemma 4.1,
$M^{*}\in\mathcal{WI}_{k}^{n}(R)^{\bot}$ and is also $n$-weak injective by [1,
Proposition 2.6]. Similar to the proof of $(5)\Rightarrow(4)$, we see that
$M^{*}\in\mathcal{WI}_{k}^{n}(R)^{\bot}$ and is also injective. Therefore by
Lemma 4.1 again, $M\in\mathcal{WI}_{k}^{n}(R)^{\top}$ and is also flat.

$(12)\Rightarrow(1)$ Every FP-injective $R$-module is $n$-weak injective. So,
if $\mathscr{FI_{n}}_{k}$ denotes the class of FP-injective right $R$-modules
with dimension at most $k$, then
${\mathscr{FI_{n}}_{k}}^{\top}\subseteq\mathcal{WI}_{k}^{n}(R)^{\top}$. By
assumption every right $R$-module in ${\mathscr{FI_{n}}_{k}}^{\top}$ is flat.
Hence by [14, Theorem 6.4], $wD(R)\leq k$, and then by Proposition 2.4, ${\rm
l.n.sp.gldim}(R)<\infty$.

$(6)\Rightarrow(12)$ Follows from Lemma 4.3. ∎

######  Theorem 4.5.

Let $n$ and $k$ be non-negative integers and consider the following
assertions:

  1. (1)

${\rm l.n.sp.gldim}(R)\leq k$.

  2. (2)

Every $(n,k)$-weak cotorsion right $R$-module is injective.

  3. (3)

${\rm id}_{R}(M)\leq k$ for every $(n,0)$-weak cotorsion right $R$-module $M$.

  4. (4)

Every $(n,k)$-weak cotorsion right $R$-module is in
$\mathcal{WF}_{k}^{n}(R^{op})$.

  5. (5)

${\rm id}_{R}(M)\leq k^{{}^{\prime}}$ for any $k^{{}^{\prime}}$ with $0\leq
k^{{}^{\prime}}\leq k$ and any $(n,k-k^{{}^{\prime}})$-weak cotorsion right
$R$-module $M$.

  6. (6)

${\rm id}_{R}(M)\leq k^{{}^{\prime}}$ for some $k^{{}^{\prime}}$ with $0\leq
k^{{}^{\prime}}\leq k$ and any $(n,k-k^{{}^{\prime}})$-weak cotorsion right
$R$-modul $M$.

  7. (7)

${\rm Ext}_{R}^{1+i}(M,N)=0$ for any $i\geq m$ and any $(n,k+m)$-weak
cotorsion module $M$ and $(n,k)$-weak cotorsion module $N$.

  8. (8)

Every right $R$-module $M$ has a $\mathcal{WF}_{k}^{n}(R^{op})$-cover with the
unique mapping property.

Then,
$(1)\Leftrightarrow(2)\Leftrightarrow(4)\Leftrightarrow(7)\Leftrightarrow(8)$
and $(1)\Rightarrow(5)\Rightarrow(6)\Rightarrow(3)$.

###### Proof.

$(1)\Leftrightarrow(2)$ ${\rm l.n.sp.gldim}(R)\leq k$ if and only if ${\rm
pd}_{R}(K_{n-1})\leq k$ for all special super finitely presented module
$K_{n-1}$ of any $n$-special super finitely presented left $R$-module $U$, if
and only if every right $R$-module is in $\mathcal{WF}_{k}^{n}(R^{op})$ by [1,
Proposition 3.5], if and only if ${\rm Ext}_{R}^{1}(N,C)=0$ for any $R$-module
$N$ and any $(n,k)$-weak cotorsion right $R$-module $C$.

$(1)\Rightarrow(4)$ Similar to the proof of $(1)\Rightarrow(2)$, it follows
that every right $R$-module is in $\mathcal{WF}_{k}^{n}(R^{op})$.

$(4)\Rightarrow(1)$ Let $N$ be a right $R$-module. Then by [1, Proposition
4.13 ], there exists an exact sequence $0\rightarrow N\rightarrow X\rightarrow
D\rightarrow 0$, where $X$ is in
$\mathcal{WF}_{k}^{n}(R^{op})^{\bot}=\mathcal{WC}_{k}^{n}(R^{op})$ and $D$ is
in $\mathcal{WF}_{k}^{n}(R^{op})$. By assumption, $X$ is in
$\mathcal{WF}_{k}^{n}(R^{op})$. Therefore, from [1, Corollary 34],
$N\in\mathcal{WF}_{k}^{n}(R^{op})$, and so by [1, Proposition 3.5], it follows
that ${\rm pd}_{R}(K_{n-1})\leq k$ for all special super finitely presented
module $K_{n-1}$ of any $n$-special super finitely presented left $R$-module
$U$. Consequently ${\rm l.n.sp.gldim}(R)\leq k$.

$(4)\Rightarrow(7)$ ${\rm Ext}_{R}^{1}(M^{{}^{\prime}},N)=0$ for all
$(n,k)$-weak cotorsion right $R$-modules $M^{{}^{\prime}}$ and $N$ by
assumption. Let $N$ be an $(n,k)$-weak cotorsion right $R$-module and $D^{m}$
the $m$th cosyzygy of $N$. Then, we have ${\rm
Ext}_{R}^{m+1}(M^{{}^{\prime}},N)\cong{\rm
Ext}_{R}^{1}(M^{{}^{\prime}},D^{m})$. By Proposition 3.9(2), $D^{m}$ is
$(n,k+m)$-weak cotorsion, and also it easy to see that $D^{m}$ is $(n,k)$-weak
cotorsion. Since $M^{{}^{\prime}}\in\mathcal{WF}_{k}^{n}(R^{op})$, we deduce
that ${\rm Ext}_{R}^{1}(M^{{}^{\prime}},D^{m})=0$, and hence ${\rm
Ext}_{R}^{1+i}(M^{{}^{\prime}},N)=0$ for any $i\geq m$. On the other hand,
every $(n,k+m)$-weak cotorsion $M$ is $(n,k)$-weak cotorsion. Therefore, ${\rm
Ext}_{R}^{1+i}(M,N)=0$ for any $i\geq m$ and for all $(n,k+m)$-weak cotorsion
modules $M$ and $(n,k)$-weak cotorsion right $R$-modules $N$.

$(7)\Rightarrow(4)$ By assumption, ${\rm Ext}_{R}^{1}(M,N)=0$ for all
$(n,k)$-weak cotorsion right $R$-modules $M$ and $N$. So every $(n,k)$-weak
cotorsion right $R$-module is in $\mathcal{WF}_{k}^{n}(R^{op})$.

$(1)\Rightarrow(8)$ Clear.

$(8)\Rightarrow(4)$ Let $M$ be an $(n,k)$-weak cotorsion right $R$-module.
Then by assumption, we have the following commutative diagram

|
---|---
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{0}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\scriptstyle{\gamma\beta}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\textstyle{0,}$$\textstyle{X^{{}^{\prime}}}$

where $\alpha$ and $\gamma$ are $\mathcal{WF}_{k}^{n}(R^{op})$-envelope. Since
$\gamma\beta\alpha=0$, it follows that $\gamma\beta=0$ by assumption.
Consequently $L={\rm im}(\beta)\subseteq{\rm ker}(\gamma)=0$ and so $L=0$.
Hence, $M\cong X$, and so $M$ is in $\mathcal{WF}_{k}^{n}(R^{op})$.

$(1)\Rightarrow(5)$ Let $N$ be a right $R$-module. By [1, Proposition 3.3],
$n.{\rm wfd}_{R}(N)\leq k$, and so there is an exact sequence:

$0\to F_{k}\to F_{k-1}\to\cdots\to F_{1}\to F_{0}\to N\to 0,$

where any $F_{i}$ is $n$-weak flat. Since $0\leq k^{{}^{\prime}}\leq k$, the
$(k^{{}^{\prime}}-1)$th syzygy of
$N\in\mathcal{WF}_{k-k^{{}^{\prime}}}^{n}(R^{op})$. If $L$ is the
$(k^{{}^{\prime}}-1)$th syzygy of $N$, then ${\rm
Ext}_{R}^{k^{{}^{\prime}}+1}(N,M)\cong{\rm Ext}_{R}^{1}(L,M)=0$ for any
$(n,k-k^{{}^{\prime}})$-weak cotorsion right $R$-module $M$. Hence, ${\rm
id}_{R}(M)\leq k^{{}^{\prime}}$.

$(5)\Rightarrow(6)$ Clear.

$(6)\Rightarrow(3)$ Let $M$ be an $(n,0)$-weak cotorsion right $R$-module.
Then, by Proposition 3.9(2), the $(n,k-k^{{}^{\prime}})$th cosyzygy
$D^{k-k^{{}^{\prime}}}$ of $M$ is $(n,k-k^{{}^{\prime}})$-weak cotorsion. So,
${\rm id}_{R}(D)\leq k^{{}^{\prime}}$ and hence ${\rm id}_{R}(M)\leq k$.

∎

######  Proposition 4.6.

Let $n$ and $k$ be non-negative integers. If $R$ satisfies one of the
following conditions:

  1. (1)

Every $(n,k)$-weak cotorsion right $R$-module has a
$\mathcal{WF}_{k}^{n}(R^{op})$-envelope with the unique mapping property.

  2. (2)

Every finitely presented right $R$-module has a
$\mathcal{WF}_{k}^{n}(R^{op})$-envelope with the unique mapping property.

Then, ${\rm l.n.sp.gldim}(R)\leq k+2$.

###### Proof.

Assume (1). Then, by [1, Theorem 4.5], every right $R$-module has a
$\mathcal{WF}_{k}^{n}(R^{op})$-cover. So, by Proposition 3.5(3) and [23, Lemma
2.1.1], we have exact sequences $0\to K\stackrel{{\scriptstyle\displaystyle
i}}{{\to}}F\stackrel{{\scriptstyle\displaystyle\alpha}}{{\to}}N\to 0$ and
$0\to K^{{}^{\prime}}\stackrel{{\scriptstyle\displaystyle
i^{{}^{\prime}}}}{{\to}}F^{{}^{\prime}}\stackrel{{\scriptstyle\displaystyle\alpha^{{}^{\prime}}}}{{\to}}K\to
0,$ where $\alpha$ and $\alpha^{{}^{\prime}}$ are
$\mathcal{WF}_{k}^{n}(R^{op})$-covers of $N$ and $K$, respectively. Also $K$
and $K^{{}^{\prime}}$ are $(n,k)$-weak cotorsion. Hence, we obtain the exact
sequence:

$0\to K^{{}^{\prime}}\stackrel{{\scriptstyle\displaystyle
i^{{}^{\prime}}}}{{\to}}F^{{}^{\prime}}\stackrel{{\scriptstyle\displaystyle
i{\alpha}^{{}^{\prime}}}}{{\to}}F\stackrel{{\scriptstyle\displaystyle\alpha}}{{\to}}N\to
0.$

Let $\beta:K^{{}^{\prime}}\to L$ be a $\mathcal{WF}_{k}^{n}(R^{op})$-envelope
with the unique mapping property. Then there exists $\gamma:L\to
F^{{}^{\prime}}$ such that $i^{{}^{\prime}}=\gamma\beta$. So
$i{\alpha}^{{}^{\prime}}\gamma\beta=i{\alpha}^{{}^{\prime}}i^{{}^{\prime}}=0$,
and then $i{\alpha}^{{}^{\prime}}\gamma=0$, which implies that ${\rm
im}(\gamma)\subseteq{\rm ker}(i{\alpha}^{{}^{\prime}})={\rm
im}(i^{{}^{\prime}})$. Therefore, there exists $\delta:L\to K^{{}^{\prime}}$
such that $i^{{}^{\prime}}\delta=\gamma$. Thus, we get the following exact
commutative diagram:

---
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K^{{}^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{{}^{\prime}}}$$\scriptstyle{\beta}$$\textstyle{F^{{}^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i\alpha^{{}^{\prime}}}$$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$$\textstyle{L\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\textstyle{L\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$

where $i^{{}^{\prime}}\delta\beta=i^{{}^{\prime}}$. So
$\delta\beta=1_{K^{{}^{\prime}}}$ (because $i^{{}^{\prime}}$ is monic). Thus,
$K^{{}^{\prime}}$ is isomorphic to a direct summand of $L$, and hence by using
of [1, Proposition 2.9 ], it follows that
$K^{{}^{\prime}}\in\mathcal{WF}_{k}^{n}(R^{op})$. Therefore, we deduce that
$K\in\mathcal{WF}_{k+1}^{n}(R^{op})$ and consequently
$N\in\mathcal{WF}_{k+2}^{n}(R^{op})$. Hence, by [1, Proposition 3.3],
$n$-${\rm wfd}_{R}(N)\leq k+2$, and so [1, Proposition 3.5] implies that ${\rm
l.n.sp.gldim}_{R}(R)\leq k+2$.

For the converse, by [5, Lemma 3.2], every right $R$-module has a
$\mathcal{WF}_{k+1}^{n}(R^{op})$-envelope with the unique mapping property
(because $\mathcal{WF}_{k+1}^{n}(R^{op})$ is closed under direct limits). ∎

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