# Transit Timing Variation of XO-3b: Evidence for Tidal Evolution of Hot
Jupiter with High Eccentricity

fan yang Department of Astronomy, Beijing Normal University, Beijing 100875,
People’s Republic of China  National Astronomical Observatories, Chinese
Academy of Sciences, 20A Datun Road, Chaoyang District, Beijing 100101, China
School of Astronomy and Space Science, University of Chinese Academy of
Sciences, Beijing 100049, China
Xing Wei  Department of Astronomy, Beijing Normal University, Beijing 100875,
People’s Republic of China

###### Abstract

Observed transit timing variation (TTV) potentially reveals the period decay
caused by star-planet tidal interaction which can explain the orbital
migration of hot Jupiters. We report the TTV of XO-3b, using TESS observed
timings and archival timings. We generate a photometric pipeline to produce
light curves from raw TESS images and find the difference between our pipeline
and TESS PDC is negligible for timing analysis. TESS timing presents a shift
of 17.6 minutes (80 $\sigma$), earlier than the prediction from the previous
ephemeris. The best linear fit for all timings available gives a Bayesian
Information Criterion (BIC) value of 439. A quadratic function is a better
model with a BIC of 56. The period derivative obtained from a quadratic
function is -6.2$\times$10-9$\pm$2.9$\times$10-10 per orbit, indicating an
orbital decay timescale 1.4 Myr. We find that the orbital period decay can be
well explained by tidal interaction. The ‘modified tidal quality factor’
$Q_{p}^{\prime}$ would be 1.8$\times$104$\pm$8$\times$102 if we assume the
decay is due to the tide in the planet; whereas $Q_{*}^{\prime}$ would be
1.5$\times$105$\pm$6$\times$103 if tidal dissipation is predominantly in the
star. The precession model is another possible origin to explain the observed
TTVs. We note that the follow-up observations of occultation timing and radial
velocity monitoring are needed for fully discriminating the different models.

Exoplanet systems (484), Exoplanet astronomy (486), Transit photometry(1709),
transit timing variation method (1710)

##  1 Introduction

Transit Timing Variation (TTV) potentially provides direct observational
evidence of tidal migration which probably explains the origin of hot Jupiter,
holding two processes, i.e., the reducing of the momentum and then the energy
(Hansen, 2010; Li et al., 2010; Dawson & Johnson, 2018, and reference
therein). The orbital momentum reduces when the eccentricity ($e$) is excited
by the perturbations, e.g., planet scattering, secular interaction (Jurić &
Tremaine, 2008; Wu & Lithwick, 2011; Endl et al., 2014; Petrovich, 2015; Shara
et al., 2016). The tides dissipate the orbital energy which turns into the
heat within the host star and Jupiter (Fabrycky & Tremaine, 2007; Naoz et al.,
2011; Penev et al., 2014).

Transiting Exoplanet Survey Satellite (TESS; Ricker et al., 2015; Fausnaugh et
al., 2021) launched in 2018, provides high precision photometry for hot
Jupiter transits in the whole sky. Combining the most recent TESS result and
archival data, it is capable to monitor TTVs in need of tidal dissipation
analysis. WASP-12b has been reported with orbital decaying through TTV
monitoring in recent decade, including the newest results from TESS (Campo et
al., 2011; Albrecht et al., 2012; Patra et al., 2017; Yee et al., 2020; Turner
et al., 2021). WASP-4b shows an earlier timing than ephemeris prediction at
81.6$\pm$11.7 s which turns out to be due to the R$\o$mer effect according to
the further investigation (Bouma et al., 2019, 2020).

TTVs potentially answer the key question that if the planet migrates to the
close-orbit at the early formation due to the disk friction, or at random
stages due to the perturbation and tidal dissipation (Lin et al., 1996; Wu &
Murray, 2003; Dawson & Johnson, 2018). The latter scenario would be either
predominant or an important supplement to the hot Jupiter migration, when hot
Jupiters are discovered with a period-decaying timescale significantly shorter
than star-planet age, especially if the orbital eccentricity is high.

XO-3b is reported as a migration candidate inferred from holding a high $e$ of
0.27587${}^{+0.00071}_{-0.00067}$ and high $M_{p}$/$M_{\ast}$ of 9$\times$10-3
(Bonomo et al., 2017). Hébrard et al. (2008) obtains an obliquity of 70$\pm$15
degrees from the Rossiter-McLaughlin effect. These static system properties
are believed to indicate XO-3b should hold a period decay process and should
present observable TTVs. Interestingly, XO-3b is also reported as a candidate
of apsidal precession (Jordán & Bakos, 2008; Antoniciello et al., 2021) which
could be another possible physical contributor of observed TTVs.

In a previous work (Shan et al., 2021, hereafter Paper I), we build a sample
of 31 hot Jupiters with significant TESS timing offsets to the previous
ephemeris, using TESS Objects of Interest (TOI) Catalog (Guerrero et al.,
2021). Among the sample, XO-3b is one of the most significant offset sources
and is briefly discussed as a candidate holding period decay.

In this work, we obtain timings of XO-3b from TESS light curves, present TTV
evidence by combining the archival timings for more than ten years, and
investigate possible TTV origins. The paper is organized as follows. In
section 2, we describe the data reduction and transit fitting. In section 3,
we present the XO-3b TTV and the most likely explanation. In Section 4, we
give a brief summary and discussion.

##  2 data reduction and transit modeling

XO-3b is a hot Jupiter transiting a F5V-type star with an orbital period of
3.19 days (Winn et al., 2008). The planet has a mass of 11.70$\pm$0.42
$M_{J}$, a radius of 1.217$\pm$0.073 $R_{J}$. The host star has a mass of
1.213$\pm$0.066 $M_{\odot}$, a radius of 1.377$\pm$0.083 $R_{\odot}$ (Bonomo
et al., 2017). The planet-star system has a semi-major axis of 4.95$\pm$0.18
(in stellar radii Stassun et al., 2017).

###  2.1 TESS Light Curve

TESS processes a field of view (FOV) of 24×96 deg2 and a pixel resolution of
21 arcseconds (Ricker et al., 2015). TESS provides continuous photometry for
the same sky area in 27 days (a sector). The images are co-added to 30 minute
cadence for the full frame images (FFI) and to 2 minute cadence near targets
of interest (target pixel files; TPF). The recently released FFIs apply 10
minute cadence after the first two-year sky survey. An example image of XO-3b
is present in Figure 1. High-quality TESS photometry has been well utilized in
exoplanet researches, e.g., the atmospheric measurement, transit timing
variation (Deming et al., 2009; Placek et al., 2016; Kempton et al., 2018;
Yang et al., 2022; Bouma et al., 2020; Yang et al., 2021; Kane et al., 2021;
Turner et al., 2021).

Figure 1: TESS cut-off image of XO-3b at 14$\times$14 pixels. The red point
indicates the target position after astrometry correction, while the green
point gives the uncorrected target position in the Gaia catalog (Gaia
Collaboration et al., 2018). The blue points present the nearby source
positions within a radius of 6 TESS pixels (126 arcseconds).

We generate a photometry pipeline, to obtain light curves from TESS raw
images. The pipeline includes the modules of e.g., astrometry correction,
contamination deblending, circular aperture photometry, and light curve
detrending (details as described in Yang et al., 2022). The pipeline generates
a consistent result when compared to Pre-search Data Conditioning (PDC) result
(Stumpe et al., 2014) from the Science Processing Operations Center (SPOC)
pipeline, among the recent applications (Yang et al., 2022, 2021, 2021). The
light curve generated by our pipeline enables us to obtain the transit
parameters of the sources without PDC light curves (similar to another TESS
photometry pipeline; Feinstein et al., 2019).

XO-3b is observed by TESS from 2019-11-28 to 2019-12-23 and is available with
the PDC light curve. We generate light curves from TPF 2-minute images and
find the difference of the conjunction timings between our result and PDC
result is within 0.3 $\sigma$ (0.1 minutes). In this work, we apply the result
from PDC light curve, in the convenience of the potential follow-up
comparison.

A detrending process is needed to remove the systematics in the light curve,
including the residual instrumental effect and the potential physical effect
such as stellar variability. These systematics can be removed by a simple
linear detrending when analyzing the transit timings (details in Yang et al.,
2022, 2021). TESS continuous observation enables one to well model the
baseline near the transit. A linear function is applied when fitting the
baseline. We also try higher-order polynomial functions which give a
negligible difference to the transit timing.

Figure 2: Folded light curve of XO-3b. The blue points present TESS
observations; green points, binned data sets for clarity; red line, fitted
light curve model; yellow region, 1 $\sigma$ significant region; black
vertical line, conjunction TESS timing obtained from modeling the light curve;
green vertical line, the predicted timing from previous ephemeris (Wong et
al., 2014). The previous ephemeris prediction is 17.8 minutes later than TESS
timing. The bottom panel gives the fitting residual, showing no structure. The
residual standard deviation is 751ppm.

###  2.2 Transit Modeling

The transit timing is obtained by modelling the light curve (Mandel & Agol,
2002), using Markov chain Monte Carlo (MCMC) (Patil et al., 2010; Foreman-
Mackey et al., 2013). The MCMC chain takes 80,000 steps after the first 30,000
steps as burn-in. We fit the light curve with a ‘Keplerian orbit’, being
consistent with the reported eccentricity as high as
0.27587${}^{+0.00071}_{-0.00067}$ (Bonomo et al., 2017). The inclination ($i$)
is fixed to 79.32, according to the reference work (Stassun et al., 2017).

The free parameters include transit mid-point, the semi-major axis
($a/R_{\ast}$), the radius ratio of the planet to the host star
($R_{p}/R_{\ast}$), the quadratic limb darkening coefficients (u1 and u2), the
argument of periapsis, the longitude of the ascending node, the time of
periapse passage. The priors of the free parameters are all uniform except for
the limb darkening. The limb darkening prior is applied as a Gaussian
distribution with a $\sigma$ of 0.05 (Yang et al., 2022), centering at the
stellar model prediction (Claret, 2018). The limb darkening prior is
0.32$\pm$0.05 (u1), 0.22$\pm$0.05 (u2), interpolated based on the stellar
parameters from TESS input catalog (Stassun et al., 2019).

We fit light curves of every transit visit as well as the conjunction light
curve folded by the previous ephemeris (Wong et al., 2014). The center of
conjunction light curve among one TESS sector ($\sim$ 30 days) would be
shifted at $\sim$ 1 minute if the folding period showing any difference at
0.0002 days. The period precision we discuss in this work is at the level of
0.00001 days which would not induce any significant timing uncertainty to the
conjunction transit mid-point. This fold-and-check algorithm has been commonly
used in the analysis of binary researches (Yang et al., 2020; Pan et al.,
2021).

##  3 TTV evidence and physical explanations

###  3.1 Timing Analysis

TESS timings are obtained through modeling the light curves (as shown in
Figure 2). We list the timings for every single transit visit and the
conjunction timing in the whole TESS sector (in Table 1). Single transit
timings and conjunction timing are well consistent within 1 $\sigma$. We note
that the difference between HJD and BJD is within 4s which is beyond the
timing precision discussed in this work. We thereby do not discriminate
between HJD and BJD in timing analysis.

Table 1: XO-3b TESS timings

TESS Transit Mid-points (HJD-2457000)
---
1819.06428$\pm$0.00035 | 1822.25556$\pm$0.00034 | 1825.44732$\pm$0.00037 | 1831.83008$\pm$0.00035
1835.02191$\pm$0.00034 | 1838.21397$\pm$0.00042 | 1831.83039$\pm$0.00016a | 1819.064098$\pm$0.00028b
Transit Mid-point Variation with Respect to previous Ephemerisa [minute]
-17.54$\pm$0.50 | -17.91$\pm$0.49 | -17.57$\pm$0.53 | -18.02$\pm$0.51
-17.59$\pm$0.48 | -16.83$\pm$0.61 | -17.57$\pm$0.22a | -17.80$\pm$0.40b

Note. (a) Conjunction timing folded on the period from Wong et al. (2014). (b)
Timing from Guerrero et al. (2021).

Figure 3: Evolution of XO-3b timings. The baseline (black dash line) is set as
the linear fit to all the available timings. Black points show the archival
timings; red diamonds, TESS timings of transit visits; green diamond, TOI
reported conjunction timing. The purple solid line and green region present
the ephemeris predicted median timing and 1 $\sigma$ region from (Wong et al.,
2014); the blue solid line and yellow region, the ephemeris prediction from
(Winn et al., 2008). The red line gives the best fitted quadratic function,
indicating a constant period derivative model.

The timings (as shown in Figure 3) show a significant inconsistency at the
recent TESS observations when compared to the most recent reported ephemeris
(Wong et al., 2014). TESS conjunction timing is 17.6 minutes ($\sim$ 80
$\sigma$) earlier than the ephemeris prediction. Any linear function,
indicating a constant period model (Winn et al., 2008, 2009; Johns-Krull et
al., 2008; Wong et al., 2014; Bonomo et al., 2017, and references therein),
does not well fit the observational timings. The Bayesian Information
Criterion (BIC) value of the best linear fit is 439 (BIC details as described
in Kass & Raftery, 1995). Conversely, the BIC is 56 for a quadratic function
which means a constant period derivative model. The physical interpretation of
such a period derivative model is available in Section 3.2. The $\Delta$(BIC)
is 383, indicating that the quadratic model is significantly preferred (as
shown in Table 2).

Bayesian analysis is sensitive to the data uncertainties. Any refinement of
timing uncertainties could significantly modify the BIC values. We test the
robustness of BIC preference by amplifying the timing uncertainties by factors
of 2 and 3. The $\Delta$(BIC) is 98 when timing uncertainties are enlarged
twofold. The $\Delta$(BIC) is 47 when uncertainties increase threefold.

The accuracy of our generated TESS timing is evaluated by comparing it to the
timing reported by TESS Objects of Interest (TOI) Catalog (Guerrero et al.,
2021). TOI timing of XO-3b is highly consistent with our generated TESS
timings. The difference between the TOI timing and our conjunction timing
(when folded into the same epoch) is within 0.3 minutes (0.5 $\sigma$). In
Paper I, we compare the timings of 262 hot Jupiters listed in the TOI catalog
to the archival ephemeris (Akeson et al., 2013) and find 159 of them show
consistent results. We check our generated timings and TOI timings, finding no
significant difference (details in Paper I). XO-3b is one of three sources
presenting the largest timing offsets among which WASP-161b has been reported
with TTV evidence (Yang et al. submitted). In reference works, we also present
the comparison between our generated TESS timings and previous ephemeris. The
comparison sample includes WASP-58b and HAT-P-31b, presenting a negligible
difference.

###  3.2 Possible Physical Explanations

The star-planet tidal interaction induces the torque to transfer angular
momentum between orbit and spins and the energy dissipation inside star and
planet, such that it influences the orbital dynamics in some respects:
migration (the change in semi-major axis or equivalently orbital period),
synchronization (the change in the difference of orbital and spin
frequencies), circularization (the change in orbital eccentricity), and the
change in orbital obliquity (Goldreich & Soter, 1966; Hut, 1981; Bodenheimer
et al., 2001; Ogilvie, 2014). The time scale of circulization ($\sim$ 1 Myr)
is much smaller than that of host star evolution. In addition, high
eccentricity can be excited by some other mechanisms, e.g., planet-planet
scattering, Kozai-Lidov cycles, secular chaos (Wu & Lithwick, 2011; Naoz et
al., 2011; Dawson & Johnson, 2018). High eccentricity can induce strong tidal
dissipation which in turn induces orbital decay, that is the most plausible
explanation for the observed TTV.

The period decay model predicts a quadratic function between the transit times
and the Nth transit epoch (Patra et al., 2017):

$\displaystyle t_{\mathrm{tra}}(N)$
$\displaystyle=t_{0}+NP+\frac{1}{2}\frac{dP}{dN}N^{2}.$

$t_{\mathrm{tra}}(N)$ is the Nth transit timing, $t_{0}$ is the timing zero
point. Therefore, the TTV is a function of time as well (as shown Table 2).
Fitting the TTVs, the derived period derivative ($\dot{P}$) is
-6.2$\times$10-9$\pm$2.9$\times$10-10 days per orbit per day. The orbital
decay timescale $P$/$\dot{P}$ is 1.4 Myr.

Table 2: Models for Observed TTV Linear Model: td = al$\times$t + bl
---
Period Decay Model: td = ad$\times$t2 \+ bd$\times$t + cd
Parameters | Timing
Linear Model
al | -0.00441$\pm$6.1$\times$ 10-5
bl | -1.23$\pm$0.13
BIC | 439
Period Decay Model
ad | -1.396$\times$10-6$\pm$6.6$\times$10-8
bd | -0.0002$\pm$0.0002
cd | 0.58$\pm$0.16
$\dot{P}$ (per orbit) | -6.2$\times$10-9$\pm$2.9$\times$10-10
BIC | 56

The rate of orbital period change $\dot{P}$ can be estimated with the theory
of equilibrium tide (Hut, 1981; Leconte et al., 2010),

$\begin{split}\dot{P}=\frac{27\pi}{Q_{p}^{\prime}}\left(\frac{M_{\star}}{M_{p}}\right)\left(\frac{R_{p}}{a}\right)^{5}\left[N(e)\,x_{\mathrm{p}}\,\frac{\omega_{\mathrm{p}}}{n}-N_{a}(e)\right]\\\
+\frac{27\pi}{Q_{*}^{\prime}}\left(\frac{M_{p}}{M_{\star}}\right)\left(\frac{R_{\star}}{a}\right)^{5}\left[N(e)\,x_{\star}\,\frac{\omega_{\star}}{n}-N_{a}(e)\right],\\\
\end{split}$ (1)

where the subscript ‘*’ stands for the stellar parameters and ‘p’ for the
planet parameters, $M$ denotes mass, $R$ radius, $a$ semi-major axis, $n$ mean
motion, $\omega$ rotation rate, $x=\cos\epsilon$, where $\epsilon$ is the
obliquity, and $Q^{\prime}$ modified tidal quality factor. $N(e)$ and
$N_{a}(e)$ are determined by eccentricity,

$N(e)=\frac{1+\frac{15}{2}e^{2}+\frac{45}{8}e^{4}+\frac{5}{16}e^{6}}{(1-e^{2})^{6}}$

and

$N_{a}(e)=\frac{1+\frac{31}{2}e^{2}+\frac{255}{8}e^{4}+\frac{185}{16}e^{6}+\frac{25}{64}e^{8}}{(1-e^{2})^{15/2}}.$

According to Equation (1), $Q^{\prime}_{\star}$ can be estimated to be
1.5$\times$105$\pm$6$\times$103 with the assumption that the orbital decay
arises from the tide raised by planet on star. $Q^{\prime}_{p}$ can be
estimated to be 1.8$\times$104$\pm$8$\times$102 if the orbital decay arises
from the tide raised by star on planet. The estimation is based on these
parameters: stellar mass $M_{*}=1.213\pm 0.066M_{\odot}$, stellar radius
$R_{*}=1.377\pm 0.083R_{\odot}$, planet mass $M_{p}=11.70\pm 0.42M_{J}$,
planet radius $R_{p}=1.217\pm 0.073R_{J}$, semi-major axis $a=4.95\pm
0.18R_{*}$, eccentricity $e=0.27587^{+0.00071}_{-0.00067}$, and obliquity
$\epsilon=70\pm 15^{\circ}$ for the star and zero for the planet (Hébrard et
al., 2008; Bonomo et al., 2017; Stassun et al., 2017). We note that applying
certain parameters can enlarge the obtained value of $Q^{\prime}_{\star}$ and
$Q^{\prime}_{p}$, e.g., applying stellar obliquity of 37.3$\pm$3.7 (Winn et
al., 2009) would double $Q^{\prime}_{\star}$ value.

The system parameters imply the history of eccentricity excitation. Bonomo et
al. (2017) reports the stellar age 2.82${}^{+0.82}_{-0.58}$ Gyr and the
projected rotational velocity 18.54$\pm$0.17 km s-1. At such an age, the orbit
should have already been circularized. Moreover, a pseudo-synchronization
between stellar spin and the planet revolution has achieved, which is
indicated by the stellar projected rotational velocity and planet orbital
period (Hut, 1981; Levrard et al., 2009; Bonomo et al., 2017). We thereby set
values of $\frac{\omega_{\mathrm{p}}}{n}$ and $\frac{\omega_{\star}}{n}$
applied in Equation 1 as both 1. The large eccentricity
(0.27587${}^{+0.00071}_{-0.00067}$) is thus probably due to the recent (less
than a few Myr) perturbations (Dawson & Johnson, 2018).

XO-3b has been reported to experience tidal dissipation through the static
systematic features (Bonomo et al., 2017), e.g., its large eccentricity and
high mass ratio of the host star to the planet. Also, these static features
imply that XO-3b would hold precession (Jordán & Bakos, 2008; Antoniciello et
al., 2021), especially when the obliquity is as high as 70$\pm$15 degrees
(Hébrard et al., 2008). It is worth investigating the contribution of the
precession towards TTV.

Apsidal precession commonly affects the light curve shape more significantly
than the timing; and the transit duration variation (TDV) is a more evident
observational parameter than TTV for the precession measurement (Pál & Kocsis,
2008; Ragozzine & Wolf, 2009), except for the case when the impact parameter b
closes to 1/$\sqrt{2}$ (Equation 15 in Jordán & Bakos, 2008). We compare the
transit durations of TESS light curves and light curves from (Winn et al.,
2008), producing a TDV baseline longer than 10 years.

Figure 4: Fits to XO-3b light curve with archival data from Winn et al.
(2008). The symbols are the same as Figure 2. The standard deviation of the
residual (bottom panel) is 2207ppm.

Archival light curves from Winn et al. (2008) are from different band
observations. The transit duration should not change when combining light
curves in different bands. We note that combining light curves in different
bands might influence some transit parameters, e.g., transit depth,
ingress/egress shape. We fit the folded light curve, following the same steps
as described in Section 2.2 (in Figure 4). We fold the light curve in each
band and find that the timing durations show no evident difference to the
folded light curve in all bands. We note that combined timing duration
uncertainty obtained in each band according to error propagation law is a
little bit smaller than the uncertainty obtained in modeling the folded light
curve in combining bands. In this work, we cite the fitting uncertainty from
the all-band light curve as the parameter uncertainty.

The transit duration of the archival folded light curve (Winn et al., 2008) is
0.1230$\pm$0.0014 days. As a comparison, the transit duration of the TESS
folded light curve is 0.1225$\pm$0.0009 days. The difference ($\sim$ 1 minute)
is well within the observational uncertainties and is significantly smaller
than the TTV of $\sim$ 17.6 minutes.

However, the impact factor b for XO-3b is possibly very closed to 1/$\sqrt{2}$
which weakens TDV distinguishing evidence towards precession. Jordán & Bakos
(2008) deduce that the logarithmic derivative of transit duration has a
multiplying factor of $1-\frac{b^{2}}{1-b^{2}}$. The b for XO-3b would be
0.70$\pm$0.04 if applying orbital parameters obtained from Bonomo et al.
(2017). The TDV would be far smaller than 1 minute with such an impact factor.
We thereby conclude precession as a possible physical origin of the observed
TTV. Further determination requires more observations and is beyond the scope
of this work.

The possibility of R$\o$mer effect is rare that neither long-term radial
velocity changing nor significant evidence of stellar companion is found in
the previous investigations (Johns-Krull et al., 2008; Winn et al., 2008,
2009; Wong et al., 2014; Stassun et al., 2017; Bonomo et al., 2017, and
references therein). Bouma et al. (2020) model the relation between the
orbital period derivative and stellar acceleration. The acceleration rate
would be 212$\pm$10 m s-1 yr-1, relating the observed period derivative of
XO-3b. Wong et al. (2014) report an RV acceleration rate of 8.40$\pm$9.13 m
s-1 yr-1 by modeling the archival RV measurements in a baseline of more than
2000 days (see technical details in Knutson et al., 2014). This reported
acceleration rate is too small to generate a comparable TTV observed.

It is known that WASP-12b near Roche limit loses mass at a rate about
$10^{-7}M_{J}$ per year due to the tidal heating in the planet (Li et al.,
2010) or the interaction of the stellar wind with the planetary magnetosphere
(Lai et al., 2010). Such a large mass loss rate probably has an appreciable
impact on the orbital decay of WASP-12b. However, for XO-3b, its Roche limit
is estimated to be $\sim 0.4R_{*}$ but its periastron to be $\sim 3.4R_{*}$,
and therefore, the mass loss is unlikely to occur on this planet.

##  4 discussion and summary

We report the evidence of XO-3b TTVs, showing an earlier timing in late 2019
TESS observation at 17.6 minutes when compared to the constant period
ephemeris (Wong et al., 2014). A quadratic function, indicating a constant
period derivative, is a significantly better fit to timings available than any
constant period fit ($\Delta$(BIC) $\geq$ 383). The derived $\dot{P}$ is
-6.2$\times$10-9$\pm$2.9$\times$10-10 days per orbit per day, implying a decay
timescale $P$/$\dot{P}$ of 1.4 Myr. We note that a precession model may be
also possible though we majorly discuss the tidal dissipation model in
explaining observed TTVs.

The theory of equilibrium tide predicts $Q^{\prime}_{\star}$
1.5$\times$105$\pm$6$\times$103 or $Q^{\prime}_{p}$
1.8$\times$104$\pm$8$\times$102, which is consistent with the reported
parameters of planet-star system. The TDV should be more significant than TTV
which is not the case.

We have scheduled new transit observations in Sitian Early commissioning,
using three 30 cm prototype telescopes (Liu et al., 2021). The new transit
timings would provide further information to discriminate the difference
between precession and period decay models.

This work made use of the NASA Exoplanet Archive
111https://exoplanetarchive.ipac.caltech.edu/index.html (Akeson et al., 2013)
and PyAstronomy222https://github.com/sczesla/PyAstronomy (Czesla et al.,
2019). We would like to thank Bo Zhang for the helpful discussion. Fan Yang
acknowledge funding from the Cultivation Project for LAMOST Scientific Payoff
and Research Achievement of CAMS-CAS. Xing Wei is supported by National
Natural Science Foundation of China (NSFC; No.11872246, 12041301), and the
Beijing Natural Science Foundation (No. 1202015).

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  * November 2022

# Shape optimization of superconducting transmon qubit for low surface
dielectric loss

Sungjun Eun, Seong Hyeon Park, Kyungsik Seo, Kibum Choi and Seungyong Hahn
Department of Electrical and Computer Engineering, Seoul National University,
1 Gwanak-ro, Gwanak-gu, Seoul 08826, Republic of Korea<EMAIL_ADDRESS>and<EMAIL_ADDRESS>
###### Abstract

Surface dielectric loss of superconducting transmon qubit is believed as one
of the dominant sources of decoherence. Reducing surface dielectric loss of
superconducting qubit is known to be a great challenge for achieving high
quality factor and a long relaxation time ($T_{1}$). Changing the geometry of
capacitor pads and junction wire of transmon qubit makes it possible to
engineer the surface dielectric loss. In this paper, we present the shape
optimization approach for reducing Surface dielectric loss in transmon qubit.
The capacitor pad and junction wire of the transmon qubit are shaped as spline
curves and optimized through the combination of the finite-element method and
global optimization algorithm. Then, we compared the surface participation
ratio, which represents the portion of electric energy stored in each
dielectric layer and proportional to two-level system (TLS) loss, of optimized
structure and existing geometries to show the effectiveness of our approach.
The result suggests that the participation ratio of capacitor pad, and
junction wire can be reduced by 16% and 26% compared to previous designs
through shape optimization, while overall footprint and anharmonicity maintain
acceptable value. As a result, the TLS-limited quality factor and
corresponding $T_{1}$ were increased by approximately 21.6%.

Keywords: Shape optimization, superconducting qubits, surface dielectric loss,
transmon.

##  1 Introduction

Over the past decade, the dramatic improvement of the coherence time of
superconducting qubits [1] made the superconducting qubit a promising platform
for quantum information processing, and recent demonstration of quantum
supremacy using superconducting qubit [2] made it even more attractive. Among
various types of superconducting qubits, transmon qubit [3] is one of the most
popular structures, which is a charge qubit featuring an exponential decrease
of charge noise due to shunt capacitance connected to Josephson Junction (JJ).
One of the biggest interests of quantum engineers is the coherence of
superconducting qubit [4], which is necessary to realize practical
superconducting quantum computers. Achieving longer relaxation time ($T_{1}$)
and dephasing time ($T_{2}$) is a matter of utmost importance. It is commonly
believed that two-level system (TLS) defects [5] contained in interface layers
act as a noise source and limit quality factors in superconducting circuits
[6, 7, 8, 9, 10]. Since TLS defects are coupled to a qubit system via an
oscillating electric field in quantum circuits [11], TLS loss of transmon
qubits can be calculated from the electric field profile and geometry of the
superconducting qubit [12, 13, 14, 15, 16, 17].
One of the common approaches to reducing TLS loss is changing material
properties or fabrication methods since TLS loss is heavily dependent on
material composition [14, 18, 19, 20]. Geometry dependence of TLS loss in a
superconducting circuit was also studied in [10, 21, 15, 22], and the
potential trade-off between footprint and quality factor [20] is commonly
observed in those studies. However, to our best knowledge, shape optimization
solely focusing on reducing participation ratio and TLS loss of transmon qubit
was not vastly been studied yet. Despite the inherent uncertainty of TLS-
related effects [14, 11, 23], we can reduce TLS loss of transmon qubit with
shape optimization through proper setting of loss tangent and participation
ratio calculation.
Shape and topology optimization [24] has been studied profoundly in mechanical
engineering [25, 26, 27], antenna optimization [28, 29, 30, 31, 32, 33, 34,
35], and electromagnetic systems [24, 36]. In particular, for planar antenna,
it has been shown that essential aspects of antenna, such as impedance
matching [33], bandwidth [29, 31, 32], and current density of dipole antenna
[30], can be optimized through shape optimization. Optimization techniques
used in those studies include adjoint-based sensitivity analysis [33, 30, 37],
genetic algorithm (GA) [38, 35], particle swarm optimization (PSO) [31, 32]
and artificial neural network [39].
In this paper, we present the shape optimization-based approach to reduce
surface dielectric loss and participation of transmon qubit. Inspired by shape
optimization techniques of planar antenna, we find a geometry that can achieve
lower participation while the overall footprint is limited. Similar to spline-
based planar antenna [29, 31, 32, 35], we used a spline shape to express the
geometry of the capacitor pad and junction wire, and a global optimization
tool was utilized to find the optimal geometry. In section 2, the fundamental
scheme for participation ratio calculation and optimization is introduced. The
effect of superconductor’s surface impedance is also discussed to justify our
simulation settings. In section 3, the results of optimization including
convergence, optimized geometry, and extracted key parameters are presented.
At last, we give conclusions on this study by discussing the increase of TLS-
limited quality factor and corresponding $T_{1}$ assuming actual parameter
values.

##  2 Surface Participation Calculation and Optimization method

###  2.1 Surface Participation Ratio Calculation

Figure 1: Illustration of TLS layers in planar transmon and junction wire. (a)
Schematic of a floating transmon qubit. Two grey-colored capacitor pads are
connected with a junction wire represented in brown color. (b) Schematic of
TLS layers. The blue, red, and black regions represent the MA, MS, SA
interface dielectric layer. Dielectric layers are divided into “interior” and
“perimeter” regions with boundary at a distance $x_{0}=1\,\mu$m from the metal
edge, and the perimeter region is further divided into “accurate” region and
“diverging” region with a boundary at a distance $0.5x_{0}=0.5\,\mu$m from
metal edge. (c) Illustration of junction wire. The red box in the middle
represents JJ, brown color represents the junction wire connecting the
capacitor pad and JJ, and $r(y)$ denotes the half-width of the junction wire
at position $y$.

In low-temperature superconducting circuits, it has been acknowledged that
parasitic TLS defects contained in dielectric layers are a noise source of
resonators [7, 40] and cause decoherence of superconducting qubit [6]. Those
TLS-related effects of superconducting circuits can be studied numerically
[12, 13], and experimentally [11]. In superconducting qubits, a thin
dielectric layer is formed at MS (metal-substrate), MA (metal-air), and SA
(substrate-air) interface as depicted in figure 1-(a),(b), and TLS defects
formed in these interface dielectrics are our main interest. Typically, TLS
loss of superconducting qubit is studied with the participation ratio model
[12, 13, 14, 15, 16, 17]. In this model, TLS-limited quality factor $Q_{\rm
TLS}$ is expressed as,

$Q_{\rm TLS}^{-1}=\sum_{i}p_{i}\,\tan\,\delta_{i},$ (1)

where $p_{i}$ and $\tan\delta_{i}$ denotes the geometry-dependent
participation ratio and loss tangent of dielectric layer $i$ ($i=$ MS, MA,
SA). The participation ratio $p_{i}$ of each dielectric layer can be
numerically calculated as [12, 17],

$p_{i}=\frac{t_{i}\varepsilon_{i}/2}{W}\int_{i}dS\left|E_{i}\right|^{2},$ (2)

where $W,t_{i},\varepsilon_{i}$ denotes the total stored energy, thickness,
and dielectric constant of $i$’th dielectric layer. In (2), a uniform electric
field within a dielectric layer is assumed, and volume integral is replaced
with surface integral, which is valid since dielectric interface layers are
thin ($\sim$10 nm thickness) compared to the overall transmon geometry feature
size ($\sim$100 $\mu$m). However, calculating (2) directly from full
electromagnetic (EM) simulation is challenging due to the diverging electric
field at the metal edge [12, 41, 42, 43].
For capacitor pads, a common approach to solve this numerical issue is an
analytic approach using conformal mapping [42, 17], or using numerical
approximations [12, 13, 18], and we adopted the method of [13]. As illustrated
in figure 1-(b), we first divide the capacitor pad into two regions -
“interior” and “perimeter”- and the perimeter region is further divided into
“accurate” region and “diverging” region. By assuming that the electric field
converges effectively in a pad region distant more than 0.5 $\mu$m from the
metal edge, we can numerically calculate the participation ratio of the
interior region and accurate region with full EM simulation. In addition, the
local field profile nearby the metal edge follows local scaling depending on
film thickness [13]. Thus, we can calculate the participation of diverging
region using scaling factor $F_{i}$, which is defined as the ratio of electric
energy stored in accurate, diverging region.
Participation ratio of the junction wire was calculated from a separate
simulation that only includes the junction wire and nearby pad region,
assuming that the participation of junction wire is independent of capacitor
pad geometry. This assumption is valid if we maintain the overall dimension of
the junction wire much smaller than the capacitor pad feature size. For the
junction wire illustrated in figure 1-(c), we adopted the flat coax
approximation used in [17], which can effectively approximate the
$x$-dependence of the electric field to that of flat coax. Electric energy
stored in the upper junction wire can be calculated from flat coax
approximation as [17],

$U=t\varepsilon\int_{\rm
upper}E(y)^{2}r(y)\left\\{\ln\left(\frac{4r(y)}{t}\right)+5\right\\}dy,$ (3)

where $E(y)$ and $r(y)$ denotes the electric field and junction wire half-
width at the centerline ($x=0$ in figure 1-(c)) point with distance $y$ from
JJ, $t$ and $\varepsilon$ denotes the thickness and dielectric constant of the
dielectric layer, and $5$ in the integrand is a thickness correction factor.
Therefore we can calculate the participation ratio from the numerical solution
of the electric field along the centerline while assuming perfect conductor
boundary condition for superconducting thin films.

###  2.2 Optimization method

Figure 2: Relation of perimeter, interior participation ratio. $p_{\rm MS}$ of
perimeter, interior region of double pad geometry is monitored while varying
width, height of capacitor pad.

(a)

(b)

Figure 3: Spline-based geometry of capacitor pad and flowchart of the
optimization process. (a) For given control point positions ($P_{0}\sim
P_{4}$), capacitor pad geometry is constructed using Bspline [44]. Point
$P_{1}^{\prime},P_{2}^{\prime}$ and $P_{3}^{\prime}$ is symmetric to
$P_{1},P_{2}$ and $P_{3}$, and the distance between the ground plane and the
capacitor pad is fixed. (b) From the given footprint limit and wire length
constraint, the range of control point coordinates is determined, and DIRECT
optimizer searches design space to find a minimal participation ratio
geometry. After each DIRECT iteration, the optimizer checks if the maximum NFE
or dynamic criteria is met. As a result, optimized spline-shaped geometry and
corresponding participation ratio $p_{i}$ and TLS-limited quality factor
$Q_{\rm TLS}$ are obtained.

Roughly, shape or topology optimization techniques can be categorized into
gradient-based and derivative-free optimization methods [45]. For a gradient-
based approach, sensitivity analysis is necessary to express the gradient of
the objective function in terms of design variables, and it is commonly done
using the adjoint variable method [24, 28].
However, due to numerical reasons, we adopted DIRECT algorithm [46, 47, 48,
49] instead, which is a global optimization method capable of finding the
optimum region in design space within a few iterations compared to other
global optimization methods [47, 49]. During each DIRECT iteration with $N$
total design variables, $N$-dimensional design space is divided into
hyperrectangles, and then objective function is evaluated at the center of
each divided hyperrectangle. Then, among those hyperrectangles, a set of
“potentially optimal” samples [46] are selected and refined at the following
iterations. A detailed description of the algorithm is provided in detail in
[47]. DIRECT optimization scheme usually terminates when the maximum number of
function evaluations (NFE) is reached [46, 47, 48] or the reduction of the
objective function value becomes marginal [49].
In capacitor pad geometry optimization, the design variables are the
coordinates of control points described in figure 3. Total number of design
variables is eight, including the $x$ and $y$ coordinates of points $P_{1}$,
$P_{2}$, $P_{3}$, and the $y$ coordinates of $P_{0}$ and $P_{4}$ (the total
length of the junction wire is determined by the position of $P_{0}$). The
objective function was $p_{\rm MS}$ of the interior region. Typically, the
participation ratio of interior region and perimeter region move in same
direction for transmon qubits with large feature size, verified by separate
parametric sweep of double pad geometry in figure 2. Each data point in figure
2 represents simulated pad participation ratio of double pad geometry
generated by parametric sweep of capacitor pad dimension. From figure 2,
$p_{\rm MS}$ of interior, perimeter region moves in same direction and thus
the participation ratio of perimeter region can also be minimized by
optimizing the participation ratio of interior region for this particular
class of floating transmons. For this reason, only the participation ratio of
the interior region is included in our objective function. The total
participation ratio including the perimeter region is evaluated afterward. The
objective function was evaluated with Ansys HFSS eigenmode solver. We used
adaptive mesh techniques in which adaptive pass terminates if the difference
of $p_{\rm MS}$ between consecutive passes is less than 0.5%. The optimization
process was terminated when the NFE reached 360 iterations, or the following
conditions were satisfied.

$\displaystyle\left|\frac{f_{\rm current}-\bar{f}}{\bar{f}}\right|$
$\displaystyle\leq 0.5\cdot 10^{-2},$ (4) $\displaystyle\left|f_{\rm
current}-\bar{f}\right|$ $\displaystyle\leq 0.2\cdot 10^{-6}.$ (5)

Conditions (4), (5) are named as “dynamic criteria” in figure 3-(b). In (4)
and (5), $f_{\rm current}$ denotes the objective function value of potentially
optimal hyperrectangles of current DIRECT iteration, and $\bar{f}$ denotes the
average function value of the last three iterations. Similar to [49], this
dynamic criteria can effectively detect whether the optimization has entered
the optimum region and stop the process before it starts excessive refining of
design space. Particular limiting value of (4), (5) was selected from typical
range of adaptive pass error of finite element simulation [50] and the value
used in [49]. In addition, to maintain a typical range of anharmonicity
$\alpha\left(\approx-E_{\rm C}\right)$, $E_{\rm C}\leq 350$ MHz condition was
forced by introducing a penalty function. We used quadratic penalty function
for this constraint [45] as,

$h(\mathbf{x})=\beta\cdot\max\left(0,E_{\rm C}-E_{\rm C,thres}\right)^{2},$
(6)

where $\mathbf{x}$ denotes design variables (coordinates of control points),
$\beta$ is the adjustable constant, and $E_{\rm C,thres}$ is 350 MHz. In (6),
$E_{\rm C}$ is approximated using,

$hf_{01}\approx\sqrt{8E_{\rm J}E_{\rm C}}-E_{\rm C},$ (7)

for transmon regime $E_{\rm J}/E_{\rm C}\gg 1$ [3], and junction energy
$E_{\rm J}/h$ is assumed to be 16.35 GHz which can be calculated from $L_{\rm
J}=10$ nH assumption with the Ambegaokar-Baritoff relation [51, 52]. Using
(7), $E_{\rm C}$ was calculated from the eigenmode solution and fed into the
penalty function (6).
In addition, spline-based geometry with five control points was used to
optimize junction wire geometry. One of the control points was fixed to the
junction location, and the $y$ coordinates of the control points were fixed.
Hence there were a total of four design variables. The objective function
$p_{\rm MS}$ was calculated from the electric field along centerline $E(y)$.
During optimization, $E(y)$ was calculated using electrostatic simulation by
applying a differential voltage between the upper and lower junction wires.
Then, DIRECT optimization is again used to find a junction wire shape that
minimizes the participation ratio, using wire length obtained from capacitor
pad optimization. Detailed settings, including parameter setting and
termination criteria, are identical to capacitor pad optimization.

###  2.3 Parameter Settings and In-Depth Analysis

In our optimization, we set $\varepsilon_{i}=10$, $t_{i}=3$ nm for dielectric
interface layers, which is a simplified setting commonly used in related
studies [12, 13, 17]. For further analysis, we first calculated the TLS-
limited quality factor and corresponding $T_{1}$ value using actual material
parameters measured in the literature [14]. We compared them with existing
planar geometries of double pad capacitor design [15, 18, 19], and concentric
transmon [53]. Furthermore, we also investigated the effect of a
superconductor’s surface impedance to validate our simulation’s perfect
conductor boundary condition. Taking into account the surface impedance of
typical aluminum superconducting film, which can be obtained from the Mattis-
Bardeen kernel function [54, 55, 40], we confirmed that considering surface
impedance resulted in participation ratio and resonant frequency change less
than 0.1% at 20 mK. At the same time, it consumes significantly longer
computation time. Thus, we safely applied perfect conductor boundary condition
for superconducting films.

##  3 Results and Discussion

Figure 4: Convergence plot for capacitor pad optimization. Inset (a)-(d) shows
how the capacitor pad shape corresponds to each DIRECT iteration’s potentially
optimal hyperrectangle changes. Y-axis shows the evolution of the capacitor
pad interior $p_{\rm MS}$ value of current best sample point. Figure 5:
Capacitor pad and junction wire geometry generated by DIRECT optimization. (a)
Optimized capacitor pad geometry. The distance between the capacitor pad and
the ground plane is fixed to 100 $\mu$m, and resulting overall footprint is
763 $\mu$m $\times$ 751 $\mu$m. (b) Optimized junction wire geometry with wire
length of 81 $\mu$m and junction width of 1 $\mu$m. The red box represents JJ.
Figure 6: Examples of existing geometries. (a) Double pad capacitor geometry
[18] with an overall footprint of 800 $\mu$m $\times$ 600 $\mu$m. (b)
Concentric transmon [53] with outer diameter 800 $\mu$m.

Optimization of both capacitor pad and junction wire converged within 360
function evaluation with maximum footprint 800 $\mu$m $\times$ 800 $\mu$m.
Figure 4 shows the evolution of the participation ratio in capacitor pad
geometry optimization. In figure 4, the y-axis shows the capacitor pad
interior $p_{\rm MS}$ value of current best sample point. At the early stage
of optimization (inset a, b of figure 4), $E_{\rm C}/h$ exceeds the set upper
bound of 350 MHz. However, after a few iterations, the optimizer approaches
the design space region that satisfies $E_{\rm C}$ constraint. After 340
function evaluation, the objective function value converges, and termination
conditions are met. DIRECT optimization of junction wire geometry also showed
similar convergence and converged in 330 function evaluation.
The resulting capacitor pad and junction wire geometries are shown in figure
5-(a),(b), respectively. The optimized capacitor geometry in figure 5-(a) can
be seen as a typical double pad capacitor being smoothly tapered near the
junction wire. Similar to the tapered wire suggested in [17], optimized
junction wire geometry figure 5-(b) shows tapered wire with a taper slope
$S\approx 0.4$, with a narrow neck in the middle bringing an additional
reduction of participation.

Table 1: Key parameters of the optimized capacitor pad, junction wire
geometry, and existing structures

|  | Optimized pad | Double pad [18] | Concentric [53]
---|---|---|---|---
$p_{\rm MS}$ | [ppm] | 81.5 | 97.5 | 106.5
$p_{\rm SA}$ | [ppm] | 70.7 | 82.7 | 88.1
$p_{\rm MA}$ | [ppm] | 6.09 | 7.46 | 8.44
|  | Optimized wire | Straight | Linear taper
$p_{\rm MS}$ | [ppm] | 16.8 | 22.8 | 17.8
$p_{\rm MA}$ | [ppm] | 0.17 | 0.23 | 0.18

*$\varepsilon_{i}=10,t_{i}=3$ nm for all dielectric layers

Key parameters of the optimized capacitor pad, junction wire, and some
reference geometries are extracted from the EM simulation and summarized in
table 1. Figure 6 shows the geometry and dimension of the existing structures.
In table 1, total MS, MA participation ratio was calculated using scaling
factor $F_{i}$, which was determined by 2D local electrostatic simulation of
the thin metal film with a thickness of 0.1 $\mu$m. For double pad capacitor
geometry, we used the one in [18], and concentric transmon geometry has an
outer diameter of 800 $\mu$m to match the overall footprint of our design.
Table 1 shows that the total participation ratio of MS, MA, SA was reduced by
approximately $15\sim 18$% compared to double pad capacitor geometry. The
lower part of table 1 shows extracted participation ratio of resulting
junction wire geometry, straight wire, and linear taper with taper slope
$S=0.4$. Linear taper and optimized geometry display a participation ratio of
approximately $22\sim 26$% smaller than straight wire, and the participation
ratio of optimized wire is slightly smaller than linear taper.
Based on the parameters in table 1, TLS-limited quality factor $Q_{\rm TLS}$
and corresponding TLS-limited relaxation time $T_{1}\left(=Q_{\rm
TLS}/\omega_{01}\right)$ were calculated using realistic parameters. To
demonstrate, we assumed niobium on a silicon substrate device [56]. We used
material parameters reported in [56] ($\tan\,\delta_{\rm MS}=1.3\times
10^{-3}$, $\tan\,\delta_{\rm SA}=2.1\times 10^{-3}$, $\tan\,\delta_{\rm
MA}=4.7\times 10^{-2}$) while assuming dielectric constant $\varepsilon_{\rm
MS}=11.7$, $\varepsilon_{\rm sub}=11.7$, $\varepsilon_{\rm SA}=4.2$,
$\varepsilon_{\rm MA}=33$ and dielectric layer thickness $t_{\rm MS}=2$ nm,
$t_{\rm SA}=5$ nm, $t_{\rm MA}=5$ nm. TLS-limited quality factor and $T_{1}$
of transmon qubit with double pad capacitor shape and straight wire were
calculated as 2.24$\times 10^{6}$ and 71.1 $\mu$s assuming qubit frequency
$f_{q}=5$ GHz. On the other hand, $Q_{\rm TLS}$ and $T_{1}$ of transmon qubit
with optimized capacitor pad and junction wire geometry were 2.72$\times
10^{6}$ and 86.5 $\mu$s. Hence, both TLS-limited quality factor and relaxation
time were improved by 21.6% through shape optimization.

##  4 Conclusion

We present a shape optimization-based approach to improve the $T_{1}$ of
superconducting transmon qubits. A common approach for the reduction of
surface dielectric loss is changing material or fabrication method, but we
adopted geometry optimization techniques that were commonly used in other
electromagnetic systems. The results indicate that surface dielectric loss can
be reduced by optimizing the geometry of transmon qubit. Further research of
geometry optimization-based approach of surface dielectric loss reduction is
required to design a qubit structure with smaller footprints yet maintaining
sufficient $T_{1}$. Moreover, combining geometry optimization-based approach
with recent tantalum transmon qubits [19, 18] will further improve the state-
of-the-arts transmon qubits, and studying other loss mechanisms, including
quasiparticle loss [57] will widen the understanding of the optimized
geometry. Our transmon qubit design can also be applied to realizing practical
quantum computer since longer $T_{1}$ reduces coherence limited quantum gate
error [58].

This research was supported by 2022 Student-Directed Education Regular Program
from Seoul National University and National R&D Program through the National
Research Foundation of Korea(NRF) funded by Ministry of Science and
ICT(2022M3I9A1072846) and in part by the Applied Superconductivity Center,
Electric Power Research Institute of Seoul National University.

## Conflict of interest

The authors declare no conflict of interest.

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A. A. Tuganbaev

$\aleph_{0}$-DISTRIBUTIVE MODULES AND RINGS

A.A. Tuganbaev

National Research University ”MPEI”, Moscow, Russia

Lomonosov Moscow State University, Moscow, Russia

<EMAIL_ADDRESS>
Abstract. Let $A$ be a ring with minimum condition on principal right ideals.
It is proved that $\aleph_{0}$-distributive right (left) $A$-modules coincide
with Artinian (Noetherian) right (left) $A$-modules. Rings, over which all
right modules are direct sums of $\aleph_{0}$-distributive modules coincide
with rings of finite representation type. Rings, whose right modules are
semidistributive, coincide with Kawada rings, over basis rings of which all
right modules are completely cyclic. The studies of Tuganbaev are supported by
Russian Scientific Foundation, project 22-11-00052.

Key words. $\aleph_{0}$-distributive module, semidistributive module,
distributive module, Koethe ring, Kawada ring, ring of finite representation
type.

MSC Classification. 16G60; 16D90; 16G60

##  1 Introduction

All rings are assumed to be associative and with non-zero identity element.
Modules are assumed to be unitary and, unless otherwise specified, all modules
are right modules. The words of type ”$A$ is a right Artinian ring” mean that
”the module $A_{A}$ is Artinian”. The words of type ”$A$ is an Artinian ring”
mean that ”the modules $A_{A}$ and ${}_{A}A$ are Artinian”.

A module is said to be $\aleph_{0}$-distributive if it does not have
subfactors which are infinite direct sums of isomorphic simple modules. We
note that any distributive module (i.e., a module with distributive submodule
lattice) is $\aleph_{0}$-distributive, since every distributive module does
not have subfactors which are a direct sum of two isomorphic non-zero modules,
[12]. The ring $\mathbb{Z}[x]$ is an example of a commutative
$\aleph_{0}$-distributive ring which is not a distributive ring.

Remark 1.1. It is clear that if all factor modules of the module $M$ do not
contain infinite direct sums of non-zero submodules (i.e. are finite-
dimensional), such modules are called quotient finite-dimensional modules or
q.f.d. modules), then the module $M$ is $\aleph_{0}$-distributive. Therefore,
every Noetherian, or Artinian, or uniserial module is quotient finite-
dimensional and, in particular, an $\aleph_{0}$-distributive module. (A module
is said to be uniserial if any two its (equivalently, cyclic) submodules are
comparable with respect to inclusion. A direct sum of uniserial modules is
called a serial module.)
A direct product of an infinite number of fields is a commutative
($\aleph_{0}$-)distributive ring which is not quotient finite-dimensional.
Every uniserial module is distributive. The ring of integers $\mathbb{Z}$ is a
distributive non-uniserial $\mathbb{Z}$-module.

Remark 1.2. A ring $A$ is said to be right pure-semisimple if $A$ is a right
Artinian ring such that every right $A$-module is a direct sum of finitely
generated modules.

A ring $A$ is called a ring of finite representation type if $A$ satisfies the
following equivalent conditions.

1) $A$ is a right Artinian ring which has, up to isomorphism, only finite
number of finitely generated indecomposable right modules.

2) $A$ is a left Artinian ring which has, up to isomorphism, only finite
number of finitely generated indecomposable left modules.

3) $A$ is a right and left pure-semisimple ring.

The equivalence of the above conditions 1)–3) is well-known; for example, see
[3], [1], [11], [5]. We note that the following question remains open: is it
true that every right pure-semisimple ring is left pure-semisimple?

The first three main results of this paper are the following Theorems 1.3, 1.4
and 1.5.

Theorem 1.3. Let $A$ be ring with minimum condition on principal right ideals.
The following assertions are true.

1. The $\aleph_{0}$-distributive right $A$-modules coincide with the Artinian right $A$-modules.

2. The $\aleph_{0}$-distributive left $A$-modules coincide with the Noetherian left $A$-modules.

3. The distributive left $A$-modules coincide with the completely cyclic left $A$-modules.

4. Distributive right $A$-modules coincide with completely cyclic Artinian right $A$-modules and, in particular, have composition series.

5. If the ring $A$ also satisfies the minimum condition on principal left ideals, then every $\aleph_{0}$-distributive right module is a Noetherian Artinian module.

Theorem 1.4. For a ring $A$, the following conditions are equivalent.

1) $A$ is a right $\aleph_{0}$-distributive ring with minimum condition on
principal right ideals.

2) $A$ is a right $\aleph_{0}$-distributive ring with minimum condition on
principal left ideals.

3) $A$ is a right Artinian ring.

Theorem 1.5. Every right module over the ring $A$ is a direct sum of
$\aleph_{0}$-distributive modules if and only if $A$ is a ring of finite
representation type.
Consequently, every right module over the ring $A$ is a direct sum of
distributive modules if and only if $A$ is a ring of finite representation
type and every indecomposable right $A$-module is distributive.

Remark 1.6. Two rings $A$ and $B$ are said to be Morita equivalent if the
categories of right modules over these rings are equivalent.

A ring $A$ is called a right Koethe ring if every right $A$-module is a direct
sum of cyclic modules.

A ring $A$ is called a right Kawada ring if every ring, which is Morita
equivalent to the ring $A$, is a right Koethe ring.

We note that all Koethe rings and all Kawada rings are rings of finite
representation type.

The fourth main result of this paper is Theorem 1.7 which uses the notion of
the basis ring given in Remark 4.4(b) of Section 4 below.

Theorem 1.7. For a ring $A$, the following conditions are equivalent.

1) Every right $A$-module is semidistributive.

2) $A$ is a Kawada ring with basis ring $B$ and every right $A$-module and
every right $B$-module are direct sums of of completely cyclic distributive
modules.

3) $A$ is a Kawada ring with basis ring $B$ and every indecomposable finitely
generated right $B$-module is a completely cyclic module.

Remark 1.8. Let $A$ be a $F$-finite-dimensional algebra over a field $F$.
Kawada’s papers [7], [9], [9] contain 19 rather complex conditions on the
local idempotents of the basis ring $B$ of the algebra $A$, the fulfillment of
which is equivalent to the fact that $A$ is a right Koethe ring. In [10], this
Kawada’s result is analyzed and commented. To these 19 conditions, we can add
the following condition 20: for every local idempotent $e$ of the algebra $B$,
the module $eB_{B}$ is completely cyclic. Then one obtains a formal
description of finite-dimensional algebras over a field which are rings, over
which all modules are semidistributive. Of course, such description is not
very useful.

##  2 The proof of Theorems 1.3 and 1.4

For the convenience of the reader, we present some well-known information.

We denote by $J(M)$, the Jacobson radical of the module $M$; we assume that
$M=J(M)$ if the module $M$ does not have maximal submodules

A nonzero-left module is said to be simple if it coincides with any its non-
zero submodule. Direct sums of simple modules are called semisimple modules.
The semisimple modules coincide with non-zero modules, in which all submodules
are direct summands. The right (left) semisimple rings coincide with rings
which are are isomorphic to finite direct products of matrix rings over
division rings.

A module $M$ is said to be Artinian (resp., Noetherian) if $M$ does not
contain infinite properly descending (resp., properly ascending) chains of
submodules. A module is a Noetherian if and only if all its submodules are
finitely generated. Every right Artinian ring is right Noetherian. Every
quasi-cyclic Abelian group is an Artinian $\mathbb{Z}$-module which is not
Noetherian. If $M$ is a semisimple module, then the module $M$ is finitely
generated if and only if $M$ is a Noetherian Artinian module.

A module $M$ is called a completely cyclic module (resp., a Bezout module), if
all submodules (resp., all finitely generated submodules) of $M$ are cyclic.
Every uniserial module is a distributive Bezout module. The ring of integers
is a commutative distributive non-uniserial Besout ring. Any completely cyclic
modules is Noetherian. The polynomial ring $\mathbb{Z}[x]$ over the ring of
integers $\mathbb{Z}$ is a Noetherian $\mathbb{Z}[x]$-module which is not
completely cyclic.

A non-zero module is said to be semi-Artinian if every its non-zero factor
module contains a simple submodule. Every Artinian module is semi-Artinian.
Direct sums of an infinite number of simple modules are semi-Artinian non-
Artinian modules.

Remark 2.1. A module $M$ is said to be semilocal if $M$ is finitely generated
and the factor module $M/J(M)$ with respect to its Jacobson radical is
semisimple.

A ring $A$ is said to be semilocal if $A$ satisfies the following equivalent
conditions.

  * •

$A_{A}$ is a semilocal module.

  * •

${}_{A}A$ is a semilocal module.

  * •

The factor ring $A/J(A)$ is isomorphic to a finite direct product of matrix
rings over division rings.

We note that every local ring is semilocal, and the direct product of two
division rings is semilocal but is not local.

A ring $A$ is said to be semiprimary if $A$ is semilocal and its Jacobson
radical $J(A)$ is nilpotent. Every right or left Artinian ring is semiprimary.
If $\mathbb{R}$ and $\mathbb{Q}$ are the fields of real numbers and rational
numbers, correspondingly, then the ring of matrices of the form

$\begin{pmatrix}a&c\\\ 0&b\end{pmatrix},\quad
a,b\in\mathbb{Q},\;c\in\mathbb{Q},$

is semiprimary but is not a right or left Artinian ring.

Proposition 2.2. Let $M$ be a $\aleph_{0}$-distributive module over the ring
$A$ and the set of all non-isomorphic simple subfactors of the module $M$
finitely (for example, this is the case if the ring $A$ is semilocal). Then
$M$ is a quotient finite-dimensional module.

Proof. Since all factor modules of $\aleph_{0}$-distributive modules are
$\aleph_{0}$-distributive, it is sufficient to prove that $M$ is a finite-
dimensional module. We assume that this is not true. Then $M$ contains a
direct sum $\oplus_{k=1}^{\infty}X_{k}$ non-zero cyclic submodules $X_{k}$.
Therefore, some factor module $\overline{M}$ of the module $M$ contains the
direct sum $\oplus_{k=1}^{\infty}\overline{X_{k}}$ non-zero simple submodules
$\overline{X_{k}}$. Since the set of all non-isomorphic simple subfactors of
the module $M$ finitely, the set of simple modules $\overline{X_{k}}$ contains
an infinite number of isomorphic modules. This This is a contradiction, since
$M$ is an $\aleph_{0}$-distributive module. $\square$

Proposition 2.3. Let $M$ be a $\aleph_{0}$-distributive semi-Artinian module
over the ring $A$ and the set of all non-isomorphic simple subfactors of the
module $M$ is finite (for example, this is the case if the ring $A$ is
semilocal). Then $M$ is an Artinian module.

Proof. We assume the contrary. Then there exists an infinite properly
descending chain $X_{1}\supset X_{2}\supset\ldots$ of submodules of the module
$M$. We set $X=\cap_{k=1}^{\infty}X_{k}$, $\overline{M}=M/X$ and
$\overline{X_{k}}=X_{k}/X$. A module $\overline{M}$ contains an infinite
properly descending chain
$\overline{X_{1}}\supset\overline{X_{2}}\supset\ldots$ of submodules which has
the zero intersection. This is a contradiction, since the semi-Artinian module
$\overline{M}$ is finite-dimensional, by Proposition 2.2; therefore, it is an
essential extension of some finite direct sum $S$ of simple modules which is
an Artinian module. $\square$

A submodule $X$ of the module $M$ is said to be small in $M$ if $X+Y\neq M$
for any proper submodule $Y$ of the module $M$.

Remark 2.4. A ring $A$ satisfies minimum condition on principal right ideals
if and only if the ring $A$ is semilocal and for any sequence
$(x_{k})_{k=1}^{\infty}\subset J(A)$, the element $x_{1}x_{2}\ldots x_{n}$ is
equal to the zero for some $n$ depending on the sequence $(x_{k})$) in general
case; [Bas60] see, also [6, Theorem 11.6.3]. Such rings are also called left
perfect rings. Under these conditions ,for every left $A$-module $M$, its
submodule $J(A)M$ is small in $M$; see [6, Theorem 11.5.7].

Remark 2.5. A submodule $M$ of the module $X$ is said to be fully invariant
submodule in $X$ if $\alpha(M)\subseteq M$ for every endomorphism $\alpha$ of
the module $X$. A module $M$ is said to be quasi-invariant if all its maximal
submodules are fully invariant in $M$.

The ideals of the ring $A$ coincide with fully invariant submodules in $A_{A}$
and with fully invariant submodules in ${}_{A}A$. Therefore, a ring is right
quasi-invariant (resp., left quasi-invariant) if and only if all its maximal
right (resp., left) ideals are ideals. It is clear that the ring $A$ is quasi-
invariant right if and only if the factor ring $A/J(A)$ is quasi-invariant
right. Consequently, the factor ring $A/J(A)$ is a direct product of division
rings, then the ring $A$ is (right and left) quasi-invariant.

Proposition 2.6; [12]. Let $M$ be a distributive module. The following
assertions are true.

1. $M$ does not have subfactors which are direct sums of two isomorphic non-zero modules.

2. A module $M$ is quasi-invariant.

3. Every submodule of the module $M$ which is a finite direct sum of cyclic of modules, is cyclic.

Proposition 2.7. If $A$ is a semilocal ring, then every finitely generated
distributive right $A$-module $M$ is a cyclic module, i.e., all distributive
right $A$-modules are Bezout modules.

Proof. Since $M/MJ(A)$ is a distributive semisimple module over the semisimple
Artinian ring $A/J(A)$, then the factor module $M/MJ(A)$ is an Artinian
semisimple module, by Proposition 2.3. Therefore, distributive module
$M/MJ(A)$ is a finite direct sum of simple of modules. By Proposition 2.6(3),
the module $M/MJ(A)$ is cyclic and the submodule $MJ(A)$ is small in $M$, by
the Nakayama lemma. Therefore, the module $M$ is cyclic. $\square$

Proposition 2.8. If $A$ is a ring with minimum condition on principal right
ideals, then every $\aleph_{0}$-distributive right $A$-module $M$ is a
Artinian module.

Proof. Since $A$ is a ring with minimum condition on principal right ideals,
it is directly verified that $M$ is a semi-Artinian module. It follows from
Proposition 2.3 that the module $M$ is Artinian. $\square$

Proposition 2.9. Let $A$ be ring with minimum condition on principal left
ideals (i.e., the ring $A$ is right perfect).

1. Every $\aleph_{0}$-distributive right $A$-module $M$ is a Noetherian module.

2. Every distributive right $A$-module $M$ is a completely cyclic Noetherian module.

Proof. 1. Since all submodules $\aleph_{0}$-distributive of modules are
$\aleph_{0}$-distributive, it is sufficient to prove that module $M$ is
finitely generated. Since $M/MJ(A)$ is an $\aleph_{0}$-distributive semisimple
module over the semisimple Artinian ring $A/J(A)$, it follows from Proposition
2.3 that $M/MJ(A)$ is an Artinian semisimple module; thefore, it is finitely
generated. In addition, it folows from Remark 2.4 that the submodule $MJ(A)$
is small in $M$, since $M$ is a right module over the right perfect ring $A$.
Therefore, the module $M$ is finitely generated.

2. Let $X$ be a submodule of the module $M$. It follows from 1 that the module $X$ is finitely generated. Since the ring $A$ is semilocal, it follows from Proposition 2.7 that the module $X$ is cyclic. $\square$

Remark 2.10. It follows from Propositions 2.8 and 2.9(1) that every
$\aleph_{0}$-distributive right module over the ring with minimum condition on
principal left ideals and with minimum condition on principal left ideals for
principal right ideals is an Artinian Noetherian module. In particular, every
$\aleph_{0}$-distributive right module over a semiprimary ring is an Artinian
Noetherian module.

Remark 2.11. The completion of the proof of Theorems 1.3 and 1.4.

Proof. Item 1 of Theorem 1.3 follows from Remark 1.1 and Proposition 2.8.

Item 2 of Theorem 1.3 follows from Remark 1.1 and Proposition 2.9(1).

Item 3 of Theorem 1.3 follows from of Theorem 1.3(2) and Proposition 2.7.

Item 4 of Theorem 1.3 follows from Theorem 1.3(1) and Proposition 2.7.

Item 5 of Theorem 1.3 follows from Theorem 1.3(1) and Theorem 1.3(2).

In Theorem 1.4, the implication 3) $\Rightarrow$ 1) follows from Remark 1.1,
the implication 3) $\Rightarrow$ 2) follows from Remark 1.1 and the property
that every right Artinian ring is right and left perfect, and the implication
1) $\Rightarrow$ 3) follows from Proposition 2.8.

We prove the implication 2) $\Rightarrow$ 3) of Theorem 1.4. By Proposition
2.9(1), the ring $A$ is right Noetherian. In addition, the right perfect ring
$A$ is semilocal and its Jacobson radical is a nil-ideal. Since any nil-ideal
of the right Noetherian ring $A$ is nilpotent, the ring $A$ is semiprimary. It
is clear that the right Noetherian semiprimary ring $A$ is right Artinian.
$\square$

##  3 The proof of Theorem 1.5

Remark 3.1. A module $M$ is said to be injective with respect to the module
$X$ or $X$-injective if for any submodule $X_{1}$ in $X$ every homomorphism
$X_{1}\to M$ is extended to homomorphism $X\to M$. A module $M$ over the ring
$A$ is said to be injective if $M$ is injective with respect to any
$A$-module. For example, over finite direct products of matrix rings over
division rings, all modules are injective. In addition, an Abelian group $M$
is an injective module over the ring of integers $\mathbb{Z}$ if and only if
$M$ is a divisible Abelian group, i.e., $M$ is a direct sum of groups,
isomorphic to the additive group $\mathbb{Q}$ of rational numbers and quasi-
cyclic groups $\mathbb{Z}(p^{\infty})$.

A module $Q$ is called an essential extension of its submodule $M$ if $M\cap
X\neq 0$ for any non-zero (cyclic) submodule $X$ of the module $Q$. In this
case, $M$ is called an essential submodule in $Q$. Every module $M$ is an
essential submodule of some injective module $Q$ which is called an injective
hull of the module $M$ and the injective hull $Q$ is unique up to isomorphism.
For example, if $\mathbb{Z}$ is the ring of integers, then the additive group
$\mathbb{Q}$ is an injective hull of the module $\mathbb{Z}_{\mathbb{Z}}$.

Proposition 3.2. For any ring $A$, there exists a cardinal number $\aleph$
such that the cardinality of any non-zero $\aleph_{0}$-distributive right
$A$-module $M$ does not exceed $\aleph$.

Proof. For any cyclic right $A$-module $X$, there exists a right ideal $B$ of
the ring $A$ with $X\cong A_{A}/B$. Therefore, we can consider the set
$\\{X_{i}\\}_{i\in I}$ of all pairwise non-isomorphic cyclic right
$A$-modules. Let $Y_{i}$ be the direct sum of the countable set of isomorphic
copies of the module $X_{i}$, $i\in I$. Let $Q$ be the injective hull of the
module $\oplus_{i\in I}Y_{i}$ and let $\aleph$ be the the cardinality of the
module $Q$. With the use of the Zorn lemma, we can verify that the module $M$
is an essential extension of some submodule $\oplus_{j\in J}M_{j}$, where all
$M_{j}$ are non-zero cyclic modules. Since the module $M$ is
$\aleph_{0}$-distributive, there exists a monomorphism $f\colon\oplus_{j\in
J}M_{j}\to\oplus_{i\in I}Y_{i}$. Since the module $Q$ is injective, the
monomorphism $f$ is extended to a homomorphism $g\colon M\to Q$. In addition,
$(\oplus_{j\in J}M_{j})\cap\text{Ker }g(\oplus_{j\in J}M_{j})\cap\text{Ker
}f=0$. Therefore, $g$ is a monomorphism, since $M$ is an essential extension
of the module $\oplus_{j\in J}M_{j}$. Then module $M$ is isomorphic to a
submodule of the module $Q$. Therefore, the cardinality of the module $M$ does
not exceed $\aleph$. $\square$

Proposition 3.3; [2], also see [4, Theorem 20.23].
If there exists a cardinal number $\aleph$ such that any right $A$-module is a
direct sum of $\aleph_{0}$-generated modules, then the ring $A$ is right
Artinian.

Remark 3.4. The completion of the proof of Theorem 1.5.

Proof. If $A$ is a ring of finite representation type, then every $A$-module
is a direct sum of indecomposable finitely generated modules; these modules
are Artinian and, in particular, $\aleph_{0}$-distributive.

Now let every right module over the ring $A$ be a direct sum of
$\aleph_{0}$-distributive modules. It follows from Theorem 1.3(5) that every
$\aleph_{0}$-distributive right $A$-module is a Noetherian Artinian module.
Therefore, $A$ is a ring of finite representation type. $\square$

Proposition 3.5; [13, Theorem 3.26]. If $A$ is a right quasi-invariant ring,
then every right Bezout $A$-module is distributive.

##  4 The proof of Theorem 1.7

In Remarks 4.1, 4.2 and 4.3 below, we give some well-known information on
semiperfect rings.

Remark 4.1. A module $M$ is said to be local, if $M$ is cyclic and the factor
module $M/J(M)$ with respect to its Jacobson radical is simple.

A ring $A$ is said to be local, if $A$ satisfies the following equivalent
conditions.

1) $A_{A}$ is a local module.

2) ${}_{A}A$ is a local module.

3) The factor ring $A/J(A)$ is a division ring.

4) For any element $a\in A$, at least one of the elements $a,1-a$ is
invertible.

A non-zero idempotent $e$ of the ring $A$ is said to be local if the ring
$eAe$ is local.

Remark 4.2. Let $M$ be a distributive module over the ring $A$. If all simple
subfactors of the module $M$ are isomorphic to each other (for example, it is
the case if the ring $A$ is local), then $M$ is a uniserial module.

Proof. Indeed, if $M$ is not uniserial, then $M$ contains two cyclic
submodules $X$ and $Y$ which are incomparable with respect to inclusion. Then
the submodule $X+Y$ has a factor module $S\oplus T$, where the modules $S$ and
$T$ are simple. By assumption, the simple modules $S$ and $T$ are isomorphic
to each other; this contradicts to Proposition 2.6(1). $\square$

Remark 4.3. A ring $A$ is said to be semiperfect if $A$ satisfies the
following equivalent conditions.

1) The ring $A$ is semilocal and all idempotents of the factor ring $A/J(A)$
are lifted to idempotents of the ring $A$.

2) $A_{A}$ is a (finite) direct sum of of local modules.

3) ${}_{A}A$ is a (finite) direct sum of of local modules.

4) There exists a decomposition $1=e_{1}+\ldots+e_{n}$ of the identity element
of the ring $A$ into a sum of local pairwise orthogonal idempotents
$e_{1},\ldots,e_{n}$.

We note that if $p$ and $q$ are two distinct prime integers and the ring $A$
consists of all rational numbers such that which denominators are not divided
by $p$ or $q$ in irreducible form, then $A$ is a semilocal non-semiperfect
ring.

All semiprimary rings and serial right or left rings are semiperfect. For any
division ring $F$, the ring $F[[x,y]]$ of formal power series in two
independent variables $x$ and $y$ is semiperfect (in fact, it is local) but it
is neither a semiprimary ring, nor a right serial ring, nor a left serial
ring.

Remark 4.4. Let $A$ be a semiperfect ring.

a. The decomposition $1=e_{1}+\ldots+e_{n}$ from Remark 4.3(4) is unique up to
isomorphism, i.e., for any other decomposition $1=f_{1}+\ldots+f_{m}$ of the
identity element of the semiperfect ring $A$ into the sum of local pairwise
orthogonal idempotents $f_{1},\ldots,f_{m}$, we have that $m=n$ and there
exists a permutation $\tau$ of the set $\\{e_{1},\ldots,e_{n}\\}$ such that
$e_{i}Ae_{i}\cong f_{\tau(i)}Af_{\tau(i)}$, $e_{i}A\cong f_{\tau(i)}A$ and
$Ae_{i}\cong Af_{\tau(i)}$ for all $i\in\\{1,\ldots,n\\}$.

In addition, the sets $\\{e_{1},\ldots,e_{n}\\}$ and
$\\{f_{1},\ldots,f_{n}\\}$ contain some subsets
$\\{e_{1}^{\prime},\ldots,e_{k}^{\prime}\\}$ and
$\\{f_{1}^{\prime},\ldots,f_{\ell}^{\prime}\\}$, correspondingly, such that
$e_{i}^{\prime}A\not\cong e_{j}^{\prime}A$ and $f_{i}^{\prime}A\not\cong
f_{j}^{\prime}A$ for $i\neq j$ (equivalently, $Ae_{i}^{\prime}\not\cong
Ae_{j}^{\prime}$ and $Af_{i}^{\prime}\not\cong Af_{j}^{\prime}$ for $i\neq
j$), and $k=\ell$ and the categories of all right
$(e_{1}^{\prime}+\ldots+e_{k}^{\prime})A(e_{1}^{\prime}+\ldots+e_{k}^{\prime})$-modules
and of all right
$(f_{1}^{\prime}+\ldots+f_{k}^{\prime})A(f_{1}^{\prime}+\ldots+f_{k}^{\prime})$-modules
are equivalent to the category of all right $A$-modules (a similar assertion
is true for the corresponding categories of left modules).

b. The idempotents $e^{\prime}=e_{1}^{\prime}+\ldots+e_{k}^{\prime}$ and
$f^{\prime}=f_{1}^{\prime}+\ldots+f_{k}^{\prime}$, which are mentioned in a,
are called basis idempotents of the ring $A$, and the rings
$e^{\prime}Ae^{\prime}$ and $f^{\prime}Af^{\prime}$ are isomorphic to each
other and are called basis rings of the ring $A$. If the basis idempotent
$e^{\prime}$ is equal to of the identity element of the ring $A$ (i.e., $A$
coincides with its basis ring), then $A$ is called a basis (or self-basis)
ring.

Remark 4.5. Let $B$ be the basis ring of the semiperfect ring $A$. The
following assertions a, b and c are well-known. The assertion of d follows
from the assertions of c, Proposition 3.5 and Proposition 2.9(2). The
assertion of e follows from d.

a. The factor ring $B/J(B)$ is isomorphic to a direct product of division
rings.

b. $B$ is a basis, semiperfect, quasi-invariant ring.

c. The rings $A$ and $B$ are Morita equivalent, i.e., there exists an
equivalence functor $\mathcal{T}$ between the category of all right
$A$-modules $\text{Mod }A$ and the category of all right $B$-modules
$\text{Mod }B$ and the ring $A$ is right (resp., left) perfect if and only if
the ring $B$ is right (resp., left) perfect. In particular, distributivity
(resp., semidistributivity) of some right $A$-module $M$ if and only if the
right $B$-module $\mathcal{T}(M)$ is distributive (resp., semidistributive).

d. Distributivity (resp., semidistributivity) of some right $A$-module $M$ is
equivalent to the property that the right $B$-module $\mathcal{T}(M)$ is a
Bezout module (resp., direct sum of Bezout modules). In addition, if the ring
$A$ (and, consequently, the ring $B$) is right perfect, then distributivity
(resp., semidistributivity) of the right $A$-module $M$ is equivalent to the
property that the right $B$-module $\mathcal{T}(M)$ is completely cyclic
(resp., is a direct sum of of completely cyclic modules).

e. Every right module over the ring $A$ is a direct sum of distributive
modules if and only if $A$ is a Artinian ring of finite representation type
with basis ring $B$ and every indecomposable right $B$-module is completely
cyclic.

Remark 4.6. The completion of the proof of Theorem 1.7.

Proof. The implication 2) $\Rightarrow$ 3) is trivial.

The implication 3) $\Rightarrow$ 1) follows from Remark 4.5(e).

1) $\Rightarrow$ 2). Let every right $A$-module be semidistributive. Since the
property of semidistributivity of all right modules is preserved under Morita
equivalence, it follows from Remark 4.5(e) that $A$ is an Artinian ring of
finite representation type with basis ring $B$ and every indecomposable right
$B$-module is completely cyclic. In particular, $A$ is a right Koethe ring.
Since semidistributivity property of all right modules is preserved under
Morita equivalence, every ring, which is Morita equivalent to the ring $A$, is
a Koethe ring, i.e., the rings $A$ and $B$ are Kawada rings. $\square$

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