* Farsalinos et al., (2021) Farsalinos, K., Poulas, K., Kouretas, D., Vantarakis, A., Leotsinidis, M., Kouvelas, D., Docea, A. O., Kostoff, R., Gerotziafas, G. T., & Antoniou, M. N. (2021).  Improved strategies to counter the COVID-19 pandemic: Lockdowns vs. primary and community healthcare.  Toxicology Reports, 8, 1–9.
  * Federico & Ferrari, (2020) Federico, S. & Ferrari, G. (2020).  Taming the spread of an epidemic by lockdown policies.  Journal of Mathematical Economics, 93, 102453.
  * Fernández-Villaverde & Jones, (2020) Fernández-Villaverde, J. & Jones, C. M. (2020).  Estimating and Simulating a SIRD Model of COVID-19 for Many Countries, States, and Cities.  Working Paper 27128, National Bureau of Economic Research.
  * Forneron & Ng, (2018) Forneron, J.-J. & Ng, S. (2018).  The ABC of simulation estimation with auxiliary statistics.  Journal of Econometrics, 205(1), 112–139.
  * Frankel & Kotti, (2021) Frankel, J. A. & Kotti, R. (2021).  The Virus, Vaccination, and Voting.  Technical report, National Bureau of Economic Research.
  * Freed et al., (2020) Freed, M., Cubanski, J., Neuman, T., Kates, J., & Michaud, J. (2020).  What share of people who have died of COVID-19 are 65 and older – and how does it vary by state?  https://www.kff.org/coronavirus-covid-19/issue-brief/what-share-of-people-who-have-died-of-covid-19-are-65-and-older-and-how-does-it-vary-by-state/.  Accessed: 2021-10-12.
  * Fruchterman & Reingold, (1991) Fruchterman, T. M. & Reingold, E. M. (1991).  Graph drawing by force-directed placement.  Software: Practice and Experience, 21(11), 1129–1164.
  * Galeotti et al., (2020) Galeotti, A., Golub, B., & Goyal, S. (2020).  Targeting interventions in networks.  Econometrica, 88(6), 2445–2471.
  * Ganesh et al., (2005) Ganesh, A., Massoulié, L., & Towsley, D. (2005).  The effect of network topology on the spread of epidemics.  In Proceedings IEEE 24th Annual Joint Conference of the IEEE Computer and Communications Societies., volume 2 (pp. 1455–1466).
  * Garetto et al., (2003) Garetto, M., Gong, W., & Towsley, D. (2003).  Modeling malware spreading dynamics.  In IEEE INFOCOM 2003. Twenty-second Annual Joint Conference of the IEEE Computer and Communications Societies (IEEE Cat. No. 03CH37428), volume 3 (pp. 1869–1879).
  * Gertler & Waldman, (1992) Gertler, P. J. & Waldman, D. M. (1992).  Quality-adjusted cost functions and policy evaluation in the nursing home industry.  Journal of Political Economy, 100(6), 1232–1256.
  * (53) Gollier, C. (2020a).  Cost–benefit analysis of age-specific deconfinement strategies.  Journal of Public Economic Theory, 22(6), 1746–1771.
  * (54) Gollier, C. (2020b).  Pandemic economics: optimal dynamic confinement under uncertainty and learning.  The Geneva Risk and Insurance Review, 45(2), 80–93.
  * Green et al., (2006) Green, D. M., Kiss, I. Z., & Kao, R. R. (2006).  Parameterization of individual-based models: comparisons with deterministic mean-field models.  Journal of Theoretical Biology, 239(3), 289–297.
  * Harris, (2020) Harris, J. E. (2020).  The Subways Seeded the Massive Coronavirus Epidemic in New York City.  Working Paper 27021, National Bureau of Economic Research.
  * Hashem Pesaran & Yang, (2021) Hashem Pesaran, M. & Yang, C. F. (2021).  Matching theory and evidence on Covid-19 using a Stochastic Network SIR Model.  CESifo Working Paper, (8695).
  * Heap et al., (2020) Heap, S. P. H., Koop, C., Matakos, K., Unan, A., & Weber, N. (2020).  Valuating health vs wealth: The effect of information and how this matters for COVID-19 policymaking.  Article, VoxEU.org.
  * Hethcote, (2000) Hethcote, H. W. (2000).  The mathematics of infectious diseases.  SIAM review, 42(4), 599–653.
  * Hethcote & Yorke, (2014) Hethcote, H. W. & Yorke, J. A. (2014).  Gonorrhea transmission dynamics and control, volume 56.  Springer.
  * Hong et al., (2021) Hong, B., Bonczak, B. J., Gupta, A., Thorpe, L. E., & Kontokosta, C. E. (2021).  Exposure density and neighborhood disparities in COVID-19 infection risk.  Proceedings of the National Academy of Sciences, 118(13).
  * IMF, (2020) IMF (2020).  CHAPTER 1 THE GREAT LOCKDOWN.  World Economic Outlook.
  * Ip, (2020) Ip, G. (2020).  New thinking on Covid lockdowns: They’re overly blunt and costly.  Wall Street Journal.  Accessed: 2021-09-30.
  * Jeong et al., (2000) Jeong, H., Tombor, B., Albert, R., Oltvai, Z. N., & Barabási, A.-L. (2000).  The large-scale organization of metabolic networks.  Nature, 407(6804), 651–654.
  * John, (2020) John, M. (2020).  How is the world dealing with lockdown? This poll has captured the public’s mood.  Opinion, World Economic Forum.
  * Karaivanov, (2020) Karaivanov, A. (2020).  A social network model of covid-19.  Plos One, 15(10), e0240878.
  * Karanikolos & McKee, (2020) Karanikolos, M. & McKee, M. (2020).  How comparable is COVID-19 mortality across countries?  Eurohealth, 26(2), 45–50.
  * Keeling & Eames, (2005) Keeling, M. J. & Eames, K. T. (2005).  Networks and epidemic models.  Journal of the Royal Society Interface, 2(4), 295–307.
  * Kephart & White, (1992) Kephart, J. O. & White, S. R. (1992).  Directed-graph epidemiological models of computer viruses.  In Computation: the micro and the macro view (pp. 71–102). World Scientific.
  * Kermack & McKendrick, (1927) Kermack, W. O. & McKendrick, A. G. (1927).  A contribution to the mathematical theory of epidemics.  Proceedings of the Royal Society of London. Series A, containing papers of a mathematical and physical character, 115(772), 700–721.
  * Kermack & McKendrick, (1932) Kermack, W. O. & McKendrick, A. G. (1932).  Contributions to the mathematical theory of epidemics. ii.—the problem of endemicity.  Proceedings of the Royal Society of London. Series A, containing papers of a mathematical and physical character, 138(834), 55–83.
  * Knodel, (2020) Knodel, J. (2020).  Texas Lt. Gov. Dan Patrick suggests he, other seniors willing to die to get economy going again.  https://www.nbcnews.com/news/us-news/texas-lt-gov-dan-patrick-suggests-he-other-seniors-willing-n1167341.  Accessed: 2020-10-14.
  * Kochańczyk & Lipniacki, (2021) Kochańczyk, M. & Lipniacki, T. (2021).  Pareto-based evaluation of national responses to covid-19 pandemic shows that saving lives and protecting economy are non-trade-off objectives.  Scientific reports, 11(1), 1–9.
  * Kuchler et al., (2020) Kuchler, T., Russel, D., & Stroebel, J. (2020).  The Geographic Spread of COVID-19 Correlates with Structure of Social Networks as Measured by Facebook.  Working Paper 26990, National Bureau of Economic Research.
  * Liljeros et al., (2001) Liljeros, F., Edling, C. R., Amaral, L. A. N., Stanley, H. E., & Åberg, Y. (2001).  The web of human sexual contacts.  Nature, 411(6840), 907–908.
  * Lloyd et al., (2006) Lloyd, A. L., Valeika, S., & Cintrón-Arias, A. (2006).  Infection dynamics on small-world networks.  Contemporary Mathematics, 410, 209–234.
  * Ma et al., (2021) Ma, L., Shapira, G., De Walque, D., Do, Q.-T., Friedman, J., & Levchenko, A. A. (2021).  The Intergenerational Mortality Tradeoff of COVID-19 Lockdown Policies.  Working Paper 28925, National Bureau of Economic Research.
  * National Center for Health Statistics, (2020) National Center for Health Statistics (2020).  Provisional COVID-19 deaths by sex and age.  https://data.cdc.gov/NCHS/Provisional-COVID-19-Deaths-by-Sex-and-Age/9bhg-hcku.  Accessed: 2021-10-12.
  * Neelon et al., (2021) Neelon, B., Mutiso, F., Mueller, N. T., Pearce, J. L., & Benjamin-Neelon, S. E. (2021).  Associations between governor political affiliation and COVID-19 cases, deaths, and testing in the US.  American Journal of Preventive Medicine, 61(1), 115–119.
  * Newman, (2002) Newman, M. E. (2002).  Spread of epidemic disease on networks.  Physical Review E, 66(1), 016128.
  * Pastor-Satorras et al., (2015) Pastor-Satorras, R., Castellano, C., Van Mieghem, P., & Vespignani, A. (2015).  Epidemic processes in complex networks.  Reviews of Modern Physics, 87(3), 925.
  * Pastor-Satorras & Vespignani, (2001) Pastor-Satorras, R. & Vespignani, A. (2001).  Epidemic spreading in scale-free networks.  Physical Review Letters, 86(14), 3200.
  * Peng et al., (2020) Peng, L., Yang, W., Zhang, D., Zhuge, C., & Hong, L. (2020).  Epidemic analysis of covid-19 in china by dynamical modeling.  arXiv preprint arXiv:2002.06563.
  * Pindyck, (2020) Pindyck, R. S. (2020).  COVID-19 and the welfare effects of reducing contagion.  Working Paper 27121, National Bureau of Economic Research.
  * Pongou, (2010) Pongou, R. (2010).  The economics of fidelity in network formation.  PhD thesis, Brown University.
  * Pongou & Serrano, (2013) Pongou, R. & Serrano, R. (2013).  Fidelity networks and long-run trends in HIV/AIDS gender gaps.  American Economic Review, 103(3), 298–302.
  * Pongou & Serrano, (2016) Pongou, R. & Serrano, R. (2016).  Volume of trade and dynamic network formation in two-sided economies.  Journal of Mathematical Economics, 63, 147–163.
  * Pongou & Tondji, (2018) Pongou, R. & Tondji, J.-B. (2018).  Valuing inputs under supply uncertainty: The bayesian Shapley value.  Games and Economic Behavior, 108, 206–224.
  * Prem et al., (2020) Prem, K., Liu, Y., Russell, T. W., Kucharski, A. J., Eggo, R. M., Davies, N., Flasche, S., Clifford, S., Pearson, C. A., Munday, J. D., et al. (2020).  The effect of control strategies to reduce social mixing on outcomes of the covid-19 epidemic in wuhan, china: a modelling study.  The Lancet Public Health, 5(5), e261–e270.
  * Reyes-Santías et al., (2020) Reyes-Santías, F., Cordova-Arevalo, O., & Rivo-Lopez, E. (2020).  Using flexible regression models for calculating hospital’s production functions.  BMC Health Services Research, 20(1), 1–11.
  * Rice et al., (2021) Rice, B. L., Annapragada, A., Baker, R. E., Bruijning, M., Dotse-Gborgbortsi, W., Mensah, K., Miller, I. F., Motaze, N. V., Raherinandrasana, A., Rajeev, M., et al. (2021).  Variation in SARS-CoV-2 outbreaks across sub-Saharan Africa.  Nature Medicine, 27(3), 447–453.
  * Rodrigues, (2019) Rodrigues, F. A. (2019).  Network centrality: an introduction.  In A mathematical modeling approach from nonlinear dynamics to complex systems (pp. 177–196). Springer.
  * Schellekens & Sourrouille, (2020) Schellekens, P. & Sourrouille, D. (2020).  The unreal dichotomy in COVID-19 mortality between high-income and developing countries.  The Brookings Institution.
  * Seierstad & Sydsaeter, (1986) Seierstad, A. & Sydsaeter, K. (1986).  Optimal control theory with economic applications.  Elsevier North-Holland, Inc.
  * Smith Jr, (1993) Smith Jr, A. A. (1993).  Estimating nonlinear time-series models using simulated vector autoregressions.  Journal of Applied Econometrics, 8(S1), S63–S84.
  * Sorci et al., (2020) Sorci, G., Faivre, B., & Morand, S. (2020).  Explaining among-country variation in COVID-19 case fatality rate.  Scientific reports, 10(1), 1–11.
  * Stiglitz, (2020) Stiglitz, J. (2020).  Joseph Stiglitz on priorities for the post-COVID economy.  Opinion, World Economic Forum.
  * Van den Driessche & Watmough, (2002) Van den Driessche, P. & Watmough, J. (2002).  Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission.  Mathematical Biosciences, 180(1-2), 29–48.
  * Van Mieghem et al., (2008) Van Mieghem, P., Omic, J., & Kooij, R. (2008).  Virus spread in networks.  IEEE/ACM Transactions On Networking, 17(1), 1–14.
  * Wadhera et al., (2020) Wadhera, R. K., Wadhera, P., Gaba, P., Figueroa, J. F., Maddox, K. E. J., Yeh, R. W., & Shen, C. (2020).  Variation in COVID-19 hospitalizations and deaths across New York City boroughs.  JAMA, 323(21), 2192–2195.
  * Wallinga et al., (2006) Wallinga, J., Teunis, P., & Kretzschmar, M. (2006).  Using data on social contacts to estimate age-specific transmission parameters for respiratory-spread infectious agents.  American Journal of Epidemiology, 164(10), 936–944.
  * Wang et al., (2003) Wang, Y., Chakrabarti, D., Wang, C., & Faloutsos, C. (2003).  Epidemic spreading in real networks: An eigenvalue viewpoint.  In 22nd IEEE International Symposium on Reliable Distributed Systems, 2003. Proceedings. (pp. 25–34).
  * Watts & Strogatz, (1998) Watts, D. J. & Strogatz, S. H. (1998).  Collective dynamics of ‘small-world’ networks.  Nature, 393(6684), 440–442.
  * Wichmann & Wichmann, (2020) Wichmann, B. & Wichmann, R. (2020).  Nonparametric estimation of a primary care production function in urban brazil.  Health Economics Review, 10(1), 1–10.
  * Yang, (2021) Yang, J. (2021).  Impact of the COVID-19 pandemic on older adults in the U.S.- statistics & facts.  https://www.statista.com/topics/8276/impact-of-the-covid-19-pandemic-on-older-adults-in-the-us/.  Accessed: 2020-10-14.
  * Young, (2009) Young, H. P. (2009).  Innovation diffusion in heterogeneous populations: Contagion, social influence, and social learning.  American Economic Review, 99(5), 1899–1924.
  * Young, (2011) Young, H. P. (2011).  The dynamics of social innovation.  Proceedings of the National Academy of Sciences, 108(Supplement 4), 21285–21291.


# The rational hull of modules

gangyong lee  Gangyong Lee, Department of Mathematics Education, Chungnam
National University  Daejeon 34134, Republic of Korea  e-mail:
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we provide several new characterizations of the maximal right
ring of quotients of a ring by using the relatively dense property. As a ring
is embedded in its maximal right ring of quotients, we show that the
endomorphism ring of a module is embedded into that of the rational hull of
the module. In particular, we obtain new characterizations of rationally
complete modules. The equivalent condition for the rational hull of the direct
sum of modules to be the direct sum of the rational hulls of those modules
under certain assumption is presented. For a right $H$-module $M$ where $H$ is
a right ring of quotients of a ring $R$, we provide a sufficient condition to
be $\text{End}_{R}(M)=\text{End}_{H}(M)$. Also, we give a condition for the
maximal right ring of quotients of the endomorphism ring of a module to be the
endomorphism ring of the rational hull of a module.

###### 2020 Mathematics Subject Classification:

Primary 16D70; 16S50, Secondary 16D50

Key Words: rational hull, injective hull, maximal right ring of quotients

##  1\. Introduction

The theory of rings of quotients has its origin in the work of Ø. Ore [11] and
K. Asano [2] on the construction of the total ring of fractions, in the 1930’s
and 40’s. But the subject did not really develop until the end of the 1950’s,
when a number of important papers appeared (by R.E. Johnson [6], Y. Utumi
[15], A.W. Goldie [5], J. Lambek [8] et al). In particular, Johnson(1951),
Utumi(1956), and Findlay $\&$ Lambek(1958) have studied the maximal right ring
of quotients of a ring which is an extended ring of the base ring. For the
well-known example, the maximal right ring of quotients of integers is
rational numbers. It is the same as the injective hull of integers. For a
commutative ring, the classical right ring of quotients of a ring is to its
total quotient ring as the maximal right ring of quotients of a ring is to its
complete ring of quotients.

As we know, the study of the rational hull of a module is the same as that of
the maximal right ring of quotients in a different way. Also, like every
module has the injective hull, it is known that every module has the rational
hull in [4, Theorem 2.6]. Now, we introduce the definition of the rational
hull of a module and present its well-known results, briefly. Let $M$ be a
right $R$-module and $T=\text{End}_{R}(E(M))$. Put
$\widetilde{E}(M)=\\{x\in E(M)|\,\vartheta(M)=0~{}\text{with}~{}\vartheta\in
T~{}\Rightarrow\
\vartheta(x)=0\\}=\displaystyle\bigcap_{\begin{subarray}{c}M\subseteq\text{Ker}\vartheta\\\
\vartheta\in
T\end{subarray}}\text{Ker}\vartheta=\mathbf{r}_{E(M)}\left(\mathbf{l}_{T}(M)\right).$
Then $\widetilde{E}(M)$ is the unique maximal rational extension of $M$. We
call it the _rational hull_ of $M$. Also, it is known that
$\mathbf{r}_{E(M)}(J(T))\leq\mathbf{r}_{E(M)}\left(\mathbf{l}_{T}(M)\right)=\widetilde{E}(M)$
because $\mathbf{l}_{T}(M)\subseteq J(T)$ where $J(T)=\\{\alpha\in
T\,|\,\text{Ker}\alpha\leq^{\text{ess}}E(M)\\}$ is a Jacobson radical of a
ring $T$. Note that the maximal right ring of quotients of $R$ is
$Q(R)=\mathbf{r}_{E(R)}(\mathbf{l}_{H}(R))$ where $H=\text{End}_{R}(E(R))$
(see [8, Proposition 2]).

After the necessary background history, results, and notations in this
section, we provide several characterizations of the rational hull of a module
in Section 2 (see Theorem 2.3 and Corollary 2.10). In addition,
characterizations of rationally complete modules are presented. As a
corollary, we obtain several new characterizations of the maximal right ring
of quotients of a ring. In particular, we show that the endomorphism ring of a
module is embedded into that of the rational hull of the module as the
inherited property of its maximal right ring of quotients (see Theorem 2.15).
Our focus, in Section 3, is on the question of when is the rational hull of
the direct sum of modules the direct sum of the rational hulls of those
modules. For $M=\bigoplus_{k\in\Lambda}M_{k}$, we prove that
$\widetilde{E}(M)=\bigoplus_{k\in\Lambda}\widetilde{E}(M_{k})$ if and only if
$M_{i}$ is $M_{j}$-dense in $\widetilde{E}(M_{i})$ for all $i,j\in\Lambda$
when either $R$ is right noetherian or $|\Lambda|$ is finite (see Theorem
3.6). In the last section, we obtain a condition to be
$\text{End}_{R}(M)=\text{End}_{H}(M)$ where $H$ is a right ring of quotients
of a ring $R$ (Theorem 4.1). This condition is called the _relatively dense
property_ to a module. Also, we provide a sufficient condition for the maximal
right ring of quotients of the endomorphism ring of a module to be the
endomorphism ring of the rational hull of a module (see Theorem 4.5).

Throughout this paper, $R$ is a ring with unity and $M$ is a unital right
$R$-module. For a right $R$-module $M$, $S=\text{End}_{R}(M)$ denotes the
endomorphism ring of $M$; thus $M$ can be viewed as a left $S$\- right
$R$-bimodule. For $\varphi\in S$, $\text{Ker}\varphi$ and $\text{Im}\varphi$
stand for the kernel and the image of $\varphi$, respectively. The notations
$N\leq M$, $N\leq^{\text{ess}}M$, $N\leq^{\text{den}}M$ or $N\leq^{\oplus}M$
mean that $N$ is a submodule, an essential submodule, a dense submodule, or a
direct summand of $M$, respectively. By $E(M)$, $\widehat{M}$, and
$\widetilde{E}(M)$ we denote the injective hull, the quasi-injective hull, and
the rational hull of $M$, respectively, and $T=\text{End}_{R}(E(M))$. $Q(R)$
denotes the maximal right ring of quotients of $R$. The direct sum of
$\Lambda$ copies of $M$ is denoted by $M^{(\Lambda)}$ where $\Lambda$ is an
arbitrary index set. $\mathsf{CFM}_{\mathbb{N}}(F)$ denotes the
$\mathbb{N}\times\mathbb{N}$ column finite matrix ring over a field $F$. By
$\mathbb{Q}$, $\mathbb{Z}$, and $\mathbb{N}$ we denote the set of rational,
integer, and natural numbers, respectively. $\mathbb{Z}_{n}$ denotes the
$\mathbb{Z}$-module $\mathbb{Z}/n\mathbb{Z}$. For $x\in M$, $x^{-1}K=\\{r\in
R\,|\,xr\in K\\}\leq R_{R}$ with a right $R$-submodule $K$ of $M$. We also
denote $\mathbf{r}_{M}(I)=\\{m\in M\,|\,Im=0\\}$ for $I\leq S$ and
$\mathbf{l}_{S}(N)=\\{\varphi\in S\,|\,\varphi N=0\\}$ for $N\leq M$.

We give some properties of dense submodules. Recall that a submodule $N$ of
$M$ is said to be _dense_ in $M$ if for any $x,0\neq y\in M,$ there exists
$r\in R$ such that $xr\in N$ and $0\neq yr$.

######  Proposition 1.1 ([3, Proposition 1.3.6]).

Let $N\leq M$ be right $R$-modules. Then the following conditions are
equivalent:

  1. (a)

$N$ is dense in $M$;

  2. (b)

$\emph{Hom}_{R}(M/N,E(M))=0$;

  3. (c)

for any submodule $P$ such that $N\leq P\leq M$, $\emph{Hom}_{R}(P/N,M)=0$.

######  Proposition 1.2 ([7, Proposition 8.7]).

Let $L,N$ be submodules of a right $R$-module $M$:

  1. (i)

If $L\leq^{\emph{den}}M$ and $N\leq^{\emph{den}}M$ then $L\cap
N\leq^{\emph{den}}M$.

  2. (ii)

Let $L\leq V\leq M$. Then $L\leq^{\emph{den}}M$ if and only if
$L\leq^{\emph{den}}V$ and $V\leq^{\emph{den}}M$.

######  Proposition 1.3 ([3, Proposition 1.3.7]).

Let $M$ be a right $R$-module and $M\leq V\leq{E}(M)$. Then
$M\leq^{\emph{den}}V$ if and only if $V\leq\widetilde{E}(M)$.

We remind of some important characterizations of the rational hull of a
module.

######  Proposition 1.4.

The following hold true for a right $R$-module $M$ and
$T=\emph{End}_{R}(E(M))$:

  1. (i)

_([9, Exercises 5])_ $\widetilde{E}(M)=\\{x\in
E(M)|\,~{}\vartheta|_{M}=1_{M}~{}\emph{with}~{}\vartheta\in T\ ~{}\Rightarrow\
\vartheta(x)=x\\}$.

  2. (ii)

_([7, Proposition 8.16])_ $\widetilde{E}(M)=\\{x\in E(M)\,|\,~{}\forall y\in
E(M)\\!\setminus\\!\\{0\\},~{}y\cdot x^{-1}M\neq 0\\}$.

##  2\. The rational hull of a module

As the injective hull of a module $M$ is the minimal injective module
including $M$, the next result shows that the rational hull of a module $M$ is
the minimal rationally complete module including $M$. Recall that a right
$R$-module $M$ is said to be _rationally complete_ if it has no proper
rational (or dense) extensions, or equivalently $\widetilde{E}(M)=M$. Thus,
the rational hull $\widetilde{E}(M)$ of a module $M$ is rationally complete.

######  Theorem 2.1.

The following conditions are equivalent for right $R$-modules $M$ and $F$:

  1. (a)

$F$ is maximal dense over $M$;

  2. (b)

$F$ is rationally complete, and is dense over $M$;

  3. (c)

$F$ is minimal rationally complete, and is essential over $M$.

Note that a right $R$-module $F$ is exactly the rational hull of a module $M$
if $F$ satisfies any one of the above equivalent conditions.

###### Proof.

(a)$\Rightarrow$(b) From Proposition 1.3, it is easy to see that $F$ has no
proper dense extension. So, $F$ is a rationally complete module.
(b)$\Rightarrow$(c) Let $F^{\prime}$ be rationally complete such that $M\leq
F^{\prime}\leq F$. Since $M\leq^{\text{den}}F$, from Proposition 1.2(ii)
$M\leq^{\text{den}}F^{\prime}\leq^{\text{den}}F$. Thus, from Proposition 1.3
$F\leq^{\text{den}}\widetilde{E}(F^{\prime})=F^{\prime}$ because $F^{\prime}$
is rationally complete. Therefore $F=F^{\prime}$. (c)$\Rightarrow$(a) Let $F$
be minimal rationally complete over $M$. Since $F$ is essential over $M$,
$M\leq F\leq E(M)$. Since $M\leq^{\text{den}}\widetilde{E}(M)$,
$\text{Hom}_{R}(\widetilde{E}(M)/M,E(M))=0$. Also, since $E(F)=E(M)$,
$\text{Hom}_{R}(\widetilde{E}(M)/M,E(F))=0$. From [7, Theorem 8.24], an
inclusion map $\iota:M\rightarrow F$ extends to
$\rho:\widetilde{E}(M)\rightarrow F$ as $F$ is rationally complete (see also
Proposition 2.13). Note that $\rho$ is a monomorphism. Since
$\widetilde{E}(M)$ is rationally complete and $F$ is minimal,
$\widetilde{E}(M)=F$. ∎

The next example shows that the condition “essential over $M$” in Theorem
2.1(c) is not superfluous.

######  Example 2.2.

Let $M=\mathbb{Z}$ and $F=\mathbb{Z}_{(p)}\oplus\mathbb{Z}_{p}$ be right
$\mathbb{Z}$-modules where $\mathbb{Z}_{(p)}$ is the localization of
$\mathbb{Z}$ at the prime ideal $(p)$. It is easy to see that $M$ is not
essential in $F$, so $F$ is not a rational hull of $M$. In fact, $F$ is
minimal rationally complete over $M$: From [7, Example 8.21], $F$ is
rationally complete because $F$ is the rational hull of
$L=\mathbb{Z}\oplus\mathbb{Z}_{p}$. It is enough to show that $F$ is minimal
over $M$: Let $K$ be a rationally complete module such that $M\leq K\leq F$.
Hence $1=\text{u.dim}(M)\leq\text{u.dim}(K)\leq\text{u.dim}(F)=2$. Assume that
$\text{u.dim}(K)=1.$ Then $M\leq^{\text{ess}}K$, and hence $K$ is nonsingular
since $M$ is nonsingular. Thus $M\leq^{\text{den}}K$, which implies that
$K\cong\mathbb{Q}$ since $K$ is rationally complete and
$\widetilde{E}(M)=\mathbb{Q}$. It follows that $\mathbb{Q}$ can be embedded
into $F=\mathbb{Z}_{(p)}\oplus\mathbb{Z}_{p}$, a contradiction. Therefore,
$\text{u.dim}(K)=2.$ Then $K\leq^{\text{ess}}F$, and hence
$K\cap\mathbb{Z}_{p}\neq 0$. Thus $\mathbb{Z}_{p}\leq K$, which implies that
$L=\mathbb{Z}\oplus\mathbb{Z}_{p}\leq K$. Note that $L\leq^{\text{den}}F$
since $F=\widetilde{E}(L)$. Hence $K\leq^{\text{den}}F$, so that $K=F$ due to
the fact that $K$ is rationally complete.

We provide another characterization for the rational hull of a module using
the relatively dense property. A right ideal $I$ of a ring $R$ is called
_relatively dense to a right $R$-module_ $M$ (or $M$-_dense_) in $R$ if for
any $r\in R$ and $0\neq m\in M$, $m\cdot r^{-1}I\neq 0$. It is denoted by
$I\leq^{\text{den}}_{M}R$.

######  Theorem 2.3.

Let $M$ be a right $R$-module. Then $\widetilde{E}(M)=\\{x\in
E(M)\,|\,x^{-1}M\leq^{\emph{den}}_{M}R\\}$.

###### Proof.

Let $x\in\widetilde{E}(M)$ be arbitrary. Consider a right ideal $x^{-1}M\leq
R$. Let $0\neq m\in M$ and $r\in R$. Since
$M\leq^{\text{den}}\widetilde{E}(M)$, there exists $s\in R$ such that $ms\neq
0$ and $(xr)s=x(rs)\in M$, that is, $rs\in x^{-1}M$. Hence
$x^{-1}M\leq^{\text{den}}_{M}R$.

For the reverse inclusion, let $x\in E(M)$ such that
$x^{-1}M\leq^{\text{den}}_{M}R$. For an arbitrary nonzero element $0\neq y\in
E(M)$, it suffices to show that $y\cdot x^{-1}M\neq 0$. As
$M\leq^{\text{ess}}E(M)$, $0\neq yr\in M$ for some $r\in R$. Since
$x^{-1}M\leq^{\text{den}}_{M}R$, there exists $s\in R$ such that $yrs\neq 0$
and $rs\in x^{-1}M$. Hence $0\neq yrs\in y\cdot x^{-1}M$. Therefore
$x\in\widetilde{E}(M)$. ∎

The next definition was shown in [4, pp79] as $N\leq M(K)$, so we call a
submodule $N$ relatively dense to a module $K$ in a module $M$. (For details,
see [17].)

######  Definition 2.4.

A submodule $N$ of a right $R$-module $M$ is said to be _relatively dense to a
right $R$-module_ $K$ (or $K$-_dense_) in $M$ if for any $m\in M$ and $0\neq
x\in K$, $x\cdot m^{-1}N\neq 0$, denoted by $N\leq^{\text{den}}_{K}M$. Note
that $N$ is $M$-dense in $M$ if and only if $N$ is dense in $M$.

We provide some characterizations of the relative density property. One can
compare the following characterizations to Proposition 1.1. The equivalence
(a)$\Leftrightarrow$(c) in the following proposition is provided by [4, pp79].

######  Proposition 2.5.

The following are equivalent for right $R$-modules $M,K$ and $N\leq M$:

  1. (a)

$N$ is $K$-dense in $M$;

  2. (b)

$\emph{Hom}_{R}(M/N,E(K))=0$;

  3. (c)

for any submodule $P$ such that $N\leq P\leq M$, $\emph{Hom}_{R}(P/N,K)=0$.

###### Proof.

(a)$\Rightarrow$(b) Assume that there exists
$0\neq\alpha\in\text{Hom}_{R}(M,E(K))$ with $\alpha N=0$. Since $\alpha M\cap
K\neq 0$, there exist $x\in M$ and $0\neq y\in K$ such that $\alpha(x)=y$.
Since $N$ is $K$-dense in $M$, there exists $r\in R$ such that $xr\in N$ and
$0\neq yr$. However, $0=\alpha(xr)=\alpha(x)r=yr\neq 0$, a contradiction.
Hence $\text{Hom}_{R}(M/N,E(K))=0$.

(b)$\Rightarrow$(c) Assume that for any submodule $P$ such that $N\leq P\leq
M$, there exists $0\neq\eta\in\text{Hom}_{R}(P/N,K)$. Then by the injectivity
of $E(K)$, we can extend $\eta$ to a nonzero homomorphism from $M/N$ to
$E(K)$, a contradiction. Hence $\text{Hom}_{R}(P/N,K)=0$.

(c)$\Rightarrow$(a) Assume that $y\cdot x^{-1}N=0$ for some $x\in M$ and
$0\neq y\in K$. We define $\gamma:N+xR\rightarrow K$ given by
$\gamma(n+xr)=yr$ for $n\in N$ and $r\in R$. It is easy to see that $\gamma$
is a well-defined $R$-homomorphism vanishing on $N$. Since $N\leq N+xR\leq M$,
by hypothesis $0=\gamma(x)=y\neq 0$, a contradiction. Thus $N$ is $K$-dense in
$M$. ∎

We obtain another characterization of the relative density property related to
homomorphisms.

######  Proposition 2.6.

Let $M,K$ be right $R$-modules. Then a submodule $N$ is $K$-dense in $M$ if
and only if $\mathbf{l}_{H}(N)=0$ where $H=\emph{Hom}_{R}(M,E(K))$.

###### Proof.

Suppose $N$ is $K$-dense in $M$. Assume that $0\neq\varphi\in H$ such that
$\varphi N=0$. Then there exists $m\in M\setminus N$ such that $\varphi(m)\neq
0$. Since $\varphi(m)\in E(K)$, $0\neq\varphi(m)r\in K$ for some $r\in R$.
Hence there exists $s\in R$ such that $mrs\in N$ and $\varphi(m)rs\neq 0$
because $N\leq^{\text{den}}_{K}M$. That yields a contradiction that
$0\neq\varphi(m)rs=\varphi(mrs)\in\varphi N=0$. Therefore
$\mathbf{l}_{H}(N)=0$. Conversely, assume that $x\cdot m^{-1}N=0$ for some
$0\neq x\in K$ and $m\in M$. We define $\gamma:N+mR\rightarrow E(K)$ by
$\gamma(n+mt)=xt$ for $n\in N$ and $t\in R$. Clearly, $\gamma$ is a nonzero
$R$-homomorphism vanishing on $N$. Also, there exists
$\overline{\gamma}:M\rightarrow E(K)$ such that
$\overline{\gamma}|_{N+mR}=\gamma$. Since $0=\overline{\gamma}N$,
$0\neq\overline{\gamma}\in\mathbf{l}_{H}(N)$, a contradiction. Therefore
$x\cdot m^{-1}N\neq 0$. ∎

If $M=R$, the following result is directly provided.

######  Corollary 2.7 ([14, Proposition 1.1]).

Let $K$ be a right $R$-module and $I$ be a right ideal of a ring $R$. Then $I$
is $K$-dense in $R$ if and only if $\mathbf{l}_{E(K)}(I)=0$.

######  Proposition 2.8.

Let $K$ be a right $R$-module and $I$ be an ideal of a ring $R$. Then
$\mathbf{l}_{K}(I)=0$ if and only if $\mathbf{l}_{E(K)}(I)=0$.

###### Proof.

Since one direction is trivial, we need to show the other direction. Suppose
$\mathbf{l}_{K}(I)=0$. Assume that $\mathbf{l}_{E(K)}(I)\neq 0$. Then there
exists $0\neq x\in E(K)$ such that $xI=0$. Also, $0\neq xr\in K$ for some
$r\in R$ because $K\leq^{\text{ess}}E(K)$. Since $xrI\subseteq xI=0$, $0\neq
xr\in\mathbf{l}_{K}(I)$, a contradiction. Therefore $\mathbf{l}_{E(K)}(I)=0$.
∎

######  Corollary 2.9 ([3, Proposition 1.3.11(iv)]).

Let $I$ be an ideal of a ring $R$. Then $I\leq^{\emph{den}}R_{R}$ if and only
if $\mathbf{l}_{R}(I)=0$.

###### Proof.

The proof also follows from Corollary 2.7 and Proposition 2.8. ∎

Using Theorem 2.3 and Corollary 2.7, we obtain another characterization for
the rational hull of a module. Also, using the characterization of the
relatively dense property, new characterization for the rational hull of a
module is provided.

######  Corollary 2.10.

Let $M$ be a right $R$-module. Then the following statements hold true:

  1. (i)

_([14, Proposition 1.4(b)])_ $\widetilde{E}(M)=\\{x\in
E(M)\,|\,\mathbf{l}_{E(M)}(x^{-1}M)=0\\}$.

  2. (ii)

$\widetilde{E}(M)=\\{x\in E(M)\,|\,\emph{Hom}_{R}(R/x^{-1}M,E(M))=0\\}$.

###### Proof.

It directly follows from Theorem 2.3, Corollary 2.7, and Proposition 2.5. ∎

Several new characterizations for the maximal right ring of quotients of a
ring are provided as the following.

######  Theorem 2.11.

Let $R$ be a ring. Then the following statements hold true:

  1. (i)

A right ideal $I$ is dense in $R$ if and only if $\mathbf{l}_{E(R)}(I)=0$.

  2. (ii)

$Q(R)=\\{x\in E(R)\,|\,x^{-1}R\leq^{\emph{den}}R\\}$.

  3. (iii)

$Q(R)=\\{x\in E(R)\,|\,\mathbf{l}_{E(R)}(x^{-1}R)=0\\}$.

  4. (iv)

$Q(R)=\\{x\in E(R)\,|\,\emph{Hom}_{R}(R/x^{-1}R,E(R))=0\\}$.

We give characterizations for a rationally complete module.

######  Theorem 2.12.

The following conditions are equivalent for a right $R$-module $M$:

  1. (a)

$M$ is a rationally complete module;

  2. (b)

$\\{\overline{x}\in
E(M)/M\,|\,\mathbf{l}_{\text{E}(\text{M})}(\mathbf{r}_{\text{R}}(\overline{x}))=0\\}=\overline{0}$;

  3. (c)

For any $I\leq^{\emph{den}}_{M}R$, $\varphi\in\emph{Hom}_{R}(I,M)$ can be
uniquely extended to $\widetilde{\varphi}\in\emph{Hom}_{R}(R,M)$.

###### Proof.

Take $A:=\\{\overline{x}\in
E(M)/M\,|\,\mathbf{l}_{\emph{E}(\emph{M})}(\mathbf{r}_{\text{R}}(\overline{x}))=0\\}$.
(a)$\Rightarrow$(b) Assume that $x\in E(M)\setminus M$ such that
$\overline{x}\in A$. From Corollary 2.7,
$\mathbf{r}_{\text{R}}(\overline{x})\leq^{\text{den}}_{M}R$. Since
$\mathbf{r}_{\text{R}}(\overline{x})=x^{-1}M$,
$x^{-1}M\leq^{\text{den}}_{M}R$. Hence from Theorem 2.3
$x\in\widetilde{E}(M)=M$ because $M$ is rationally complete, a contradiction.
Therefore $A=\overline{0}$. (b)$\Rightarrow$(c) Assume to the contrary of the
condition (c). For $I\leq^{\text{den}}_{M}R$, since $M\subseteq E(M)$, there
exists $\varphi\in\text{Hom}_{R}(I,M)$ such that
$\widetilde{\varphi}\in\text{Hom}_{R}(R,E(M))$,
$\widetilde{\varphi}|_{I}=\varphi$, and $\widetilde{\varphi}(1)\notin M$.
Since $\overline{0}\neq\widetilde{\varphi}(1)+M\in E(M)/M$ and
$I\subseteq\mathbf{r}_{\text{R}}(\widetilde{\varphi}(1)+M)\leq^{\text{den}}_{M}R$,
$\mathbf{l}_{\emph{E}(\emph{M})}(\mathbf{r}_{\text{R}}(\widetilde{\varphi}(1)+M))=0$
from Corollary 2.7, a contradiction that $A=\overline{0}$. Therefore
$\varphi\in\text{Hom}_{R}(I,M)$ is extended to
$\widetilde{\varphi}\in\text{Hom}_{R}(R,M)$. For the uniqueness, the proof is
similar to that of Proposition 2.13. (c)$\Rightarrow$(a) Assume that $M$ is
not rationally complete. Then there exists $x\in\widetilde{E}(M)\setminus M$
such that $x^{-1}M\leq^{\text{den}}_{M}R$ from Theorem 2.3. Define
$\varphi:x^{-1}M\rightarrow M$ given by $\varphi(r)=xr$. By hypothesis,
$\widetilde{\varphi}(1)=x1=x\in M$, a contradiction. Therefore $M$ is
rationally complete. ∎

Next, as a ring is embedding into its maximal right ring of quotients, we
provide the relationship between the endomorphism rings of a module and its
rational hull.

######  Proposition 2.13.

Let $M$ and $K$ be right $R$-modules. For any $N\leq^{\emph{den}}_{K}M$,
$\varphi\in\emph{Hom}_{R}(N,K)$ is uniquely extended to
$\widetilde{\varphi}\in\emph{Hom}_{R}(M,\widetilde{E}(K))$ and
$\widetilde{\varphi}|_{N}=\varphi$. In addition, $\varphi
N\leq^{\emph{den}}\widetilde{\varphi}M$.

###### Proof.

(Existence) Let $\varphi\in\text{Hom}_{R}(N,K)$ be arbitrary. Then there
exists $\widetilde{\varphi}\in\text{Hom}_{R}(M,E(K))$ such that
$\widetilde{\varphi}|_{N}=\varphi$. Since $\widetilde{\varphi}$ induces a
surjection from $M/N$ to
$(\widetilde{\varphi}M+\widetilde{E}(K))/\widetilde{E}(K)$ and
$\text{Hom}_{R}(M/N,E(K))=0$ (see Proposition 2.5),
$\text{Hom}_{R}\left(\frac{\widetilde{\varphi}M+\widetilde{E}(K)}{\widetilde{E}(K)},E(K)\right)=0$.
Hence $\widetilde{E}(K)\leq^{\text{den}}\widetilde{\varphi}M+\widetilde{E}(K)$
by Proposition 1.1. As $\widetilde{E}(K)$ is rationally complete,
$\widetilde{\varphi}M\subseteq\widetilde{E}(K)$.
(Uniqueness) Suppose $\widetilde{\varphi}$ and $\widetilde{\psi}$ are in
$\text{Hom}_{R}(M,\widetilde{E}(K))$ such that
$\widetilde{\varphi}|_{N}=\widetilde{\psi}|_{N}$. It is enough to show that
$\widetilde{\varphi}=\widetilde{\psi}$. Assume that
$\widetilde{\varphi}(x)\neq\widetilde{\psi}(x)$ for some $x\in M$. Take $0\neq
y=(\widetilde{\varphi}-\widetilde{\psi})(x)\in\widetilde{E}(K)$. Thus, there
exists $r\in R$ such that $0\neq yr\in K$. Since $N\leq^{\text{den}}_{K}M$,
there exists $s\in R$ such that $xrs\in N$ and $yrs\neq 0$. This yields a
contradiction that $0\neq
yrs=(\widetilde{\varphi}-\widetilde{\psi})(xrs)=(\widetilde{\varphi}|_{N}-\widetilde{\psi}|_{N})(xrs)=0$.
Therefore $\widetilde{\varphi}=\widetilde{\psi}$.
In addition, let $x_{1}\in\widetilde{\varphi}M$ and $0\neq
x_{2}\in\widetilde{\varphi}M$. Then
$\widetilde{\varphi}(m_{1})=x_{1},\widetilde{\varphi}(m_{2})=x_{2}$ for some
$m_{1},m_{2}\in M$. As $\widetilde{\varphi}M\subseteq\widetilde{E}(K)$, $0\neq
x_{2}r\in K$ for some $r\in R$. Since $N\leq^{\text{den}}_{K}M$ and $m_{1}r\in
M$, there exists $s\in R$ such that $m_{1}rs\in N$ and $0\neq x_{2}rs$. Thus
$x_{1}rs=\widetilde{\varphi}(m_{1}rs)\in\varphi N$ and $0\neq x_{2}rs$.
Therefore $\varphi N\leq^{\text{den}}\widetilde{\varphi}M$. ∎

Note that the dense property implies the essential property, however the
relatively dense property does not imply the essential property in general:
See
$\mathbb{Z}_{p}\leq^{\text{den}}_{\mathbb{Z}}\mathbb{Z}_{p}\oplus\mathbb{Z}_{p}$
but $\mathbb{Z}_{p}\nleq^{\text{ess}}\mathbb{Z}_{p}\oplus\mathbb{Z}_{p}$ as a
$\mathbb{Z}$-module. However, Proposition 2.13 shows that $\varphi
N\leq^{\text{den}}\widetilde{\varphi}M$ when $N\leq^{\text{den}}_{K}M$ for any
$\varphi\in\text{Hom}_{R}(N,K)$. As a corollary, we have a generalized result
of Theorem 2.12((a)$\Rightarrow$(b)).

######  Corollary 2.14.

Let $M$ be a right $R$-module. If $K$ is rationally complete, then for any
$N\leq^{\emph{den}}_{K}M$, $\varphi\in\emph{Hom}_{R}(N,K)$ is uniquely
extended to $\widetilde{\varphi}\in\emph{Hom}_{R}(M,K)$ and
$\widetilde{\varphi}|_{N}=\varphi$.

######  Theorem 2.15.

Let $M$ be a right $R$-module. Then $\emph{End}_{R}(M)$ is considered as a
subring of $\emph{End}_{R}(\widetilde{E}(M))$.

###### Proof.

Since $M\leq^{\text{den}}\widetilde{E}(M)$, from Proposition 2.13
$\varphi\in\text{End}_{R}(M)$ can be uniquely extended to
$\widetilde{\varphi}\in\text{End}_{R}(\widetilde{E}(M))$ because
$\text{End}_{R}(M)\subseteq\text{Hom}_{R}(M,\widetilde{E}(M))$. Thus we have a
one-to-one correspondence between $\text{End}_{R}(M)$ and
$\\{\widetilde{\varphi}\in\text{End}_{R}(\widetilde{E}(M))\,|\,\widetilde{\varphi}|_{M}={\varphi}\in\text{End}_{R}(M)\\}$
given by $\Omega(\varphi)=\widetilde{\varphi}$. We need to check that $\Omega$
is a ring homomorphism.

(i) Since
$\Omega(\varphi+\psi)|_{M}=(\widetilde{\varphi+\psi})|_{M}=\varphi+\psi=\Omega(\varphi)|_{M}+\Omega(\psi)|_{M}=(\Omega(\varphi)+\Omega(\psi))|_{M}$,
from the uniqueness of Proposition 2.13 we have
$\Omega(\varphi+\psi)=\Omega(\varphi)+\Omega(\psi)$.

(ii) Since
$\Omega{(\varphi\circ\psi)}|_{M}=(\widetilde{\varphi\circ\psi})|_{M}=\varphi\circ\psi=\Omega(\varphi)|_{M}\circ\Omega(\psi)|_{M}=(\Omega(\varphi)\circ\Omega(\psi))|_{M}$
because $\Omega(\varphi)|_{M}\leq M$, from the uniqueness of Proposition 2.13
we have $\Omega(\varphi\circ\psi)=\Omega(\varphi)\circ\Omega(\psi)$.

Thus $\text{End}_{R}(M)$ is isomorphic to a subring of
$\text{End}_{R}(\widetilde{E}(M))$. Therefore we consider $\text{End}_{R}(M)$
as a subring of $\text{End}_{R}(\widetilde{E}(M))$. ∎

We conclude this section with results for the rational hulls of quasi-
continuous modules and quasi-injective modules.

######  Theorem 2.16.

The following statements hold true for a module $M$:

  1. (i)

If $M$ is a quasi-continuous module then $\widetilde{E}(M)$ is a quasi-
continuous module.

  2. (ii)

If $M$ is a quasi-injective module then $\widetilde{E}(M)$ is a quasi-
injective module.

###### Proof.

(i) Let $T=\text{End}_{R}(E(\widetilde{E}(M)))=\text{End}_{R}(E(M))$. From
[10, Theorem 2.8], we need to show that
$f\widetilde{E}(M)\leq\widetilde{E}(M)$ for all idempotents $f^{2}=f\in T$:
Assume that $f\widetilde{E}(M)\nleq\widetilde{E}(M)$ for some idempotent
$f^{2}=f\in T$. Then there exists $x\in\widetilde{E}(M)$ such that
$f(x)\notin\widetilde{E}(M)$. Thus, there exists $g\in T$ such that $gM=0$ and
$gf(x)\neq 0$. Since $gf(x)\in E(M)$, there exists $r\in R$ such that $0\neq
gf(xr)\in\widetilde{E}(M)$. Thus, as $M\leq^{\text{den}}\widetilde{E}(M)$ and
$xr\in\widetilde{E}(M)$, there exists $s\in R$ such that $0\neq gf(xrs)$ and
$xrs\in M$. Note that $fM\leq M$ for all idempotents $f^{2}=f\in T$ because
$M$ is quasi-continuous. However, $0\neq gf(xrs)\in gfM\leq gM=0$, a
contradiction. Therefore $\widetilde{E}(M)$ is a quasi-continuous module.

(ii) The proof is similar to that of part (i) by using [10, Corollary 1.14]. ∎

######  Remark 2.17 ([1, Theorem 5.3]).

The rational hull of every extending module is an extending module.

Note that if $M$ is an injective module then $M=\widetilde{E}(M)$ (see [7,
Examples 8.18(1)]). The next examples exhibit that the converses of Theorem
2.16 and Remark 2.17 do not hold true.

######  Example 2.18.

(i) Consider $\mathbb{Z}$ as a $\mathbb{Z}$-module. Then
$\widetilde{E}(\mathbb{Z})=\mathbb{Q}$ is (quasi-)injective, while
$\mathbb{Z}$ is not quasi-injective.

(ii)([10, Example 2.9]) Consider a ring $R=\left(\begin{smallmatrix}F&F\\\
0&F\end{smallmatrix}\right)$ where $F$ is a field. Then
$\widetilde{E}(R_{R})=\left(\begin{smallmatrix}F&F\\\
F&F\end{smallmatrix}\right)$ is injective (hence, quasi-continuous), while
$R_{R}$ is not quasi-continuous.

(iii) Consider $\mathbb{Z}^{(\mathbb{N})}$ as a $\mathbb{Z}$-module. Then
$\widetilde{E}(\mathbb{Z}^{(\mathbb{N})})=\mathbb{Q}^{(\mathbb{N})}$ is
injective (hence, extending), while $\mathbb{Z}^{(\mathbb{N})}$ is not
extending.

######  Corollary 2.19.

The maximal right ring of quotients of a quasi-continuous ring is also a
quasi-continuous ring.

######  Remark 2.20 ([7, Exercises 13.8]).

The maximal right ring of quotients of a simple (resp., prime, semiprime) ring
is also a simple (resp., prime, semiprime) ring.

######  Open Question 1.

Is the rational hull of a continuous module always a continuous module?

##  3\. Direct sum of rational hulls of modules

As we know, the injective hull of the direct sum of two modules is the direct
sum of the injective hulls of each module without any condition. However, the
rational hull case is different from the injective hull case. In this section,
we discuss the condition for the rational hull of the direct sum of two
modules to be the direct sum of the rational hulls of those modules. The next
example shows that the rational hull of the direct sum of two modules is not
the direct sum of the rational hulls of each module, in general.

######  Example 3.1.

Consider $M=\mathbb{Z}\oplus\mathbb{Z}_{p}$ as a $\mathbb{Z}$-module where $p$
is prime. Then $\widetilde{E}(\mathbb{Z})=\mathbb{Q}$ and
$\widetilde{E}(\mathbb{Z}_{p})=\mathbb{Z}_{p}$. However, by [7, Example 8.21]
$\widetilde{E}(M)=\mathbb{Z}_{(p)}\oplus\mathbb{Z}_{p}\neq\mathbb{Q}\oplus\mathbb{Z}_{p}$
where
$\mathbb{Z}_{(p)}=\\{\frac{m}{n}\in\mathbb{Q}\,|\,{m,n}\in\mathbb{Z},(n,p)=1\\}$.
Hence $M$ is not a dense submodule of $\mathbb{Q}\oplus\mathbb{Z}_{p}$: For
$(\frac{1}{p},\overline{0})$ and
$0\neq(0,\overline{1})\in\mathbb{Q}\oplus\mathbb{Z}_{p}$, there is no
$n\in\mathbb{Z}$ such that
$n(\frac{1}{p},\overline{0})\in\mathbb{Z}\oplus\mathbb{Z}_{p}$ and
$n(0,\overline{1})\neq 0$.

######  Proposition 3.2.

Let $M=\bigoplus_{k\in\Lambda}M_{k}$ where $M_{k}$ be a right $R$-module and
$\Lambda$ is any index set. If either $R$ is right noetherian or $|\Lambda|$
is finite, then
$\widetilde{E}(M)\leq\bigoplus_{k\in\Lambda}\widetilde{E}(M_{k})$.

###### Proof.

Suppose $0\neq m\in\widetilde{E}(M)$. Since $\widetilde{E}(M)\subseteq
E(M)=\oplus_{k\in\Lambda}E(M_{k})$ because $R$ is right noetherian or
$|\Lambda|$ is finite, there exists $\ell\in\mathbb{N}$ such that
$m\in\oplus_{i=1}^{\ell}E(M_{i})$. Thus, $m=(m_{1},\dots,m_{\ell})$ where
$m_{i}\in E(M_{i})$. Since $(0,\dots,0,y_{i},0,\dots,0)\cdot m^{-1}M\neq 0$
for all $0\neq y_{i}\in E(M_{i})$ and $m^{-1}M=m_{1}^{-1}M_{1}\cap\cdots\cap
m_{\ell}^{-1}M_{\ell}$, $y_{i}\cdot m_{i}^{-1}M_{i}\neq 0$ for all $0\neq
y_{i}\in E(M_{i})$. Thus, $m_{i}\in\widetilde{E}(M_{i})$ for all $1\leq
i\leq\ell$ from Proposition 1.4. So,
$m=(m_{1},\dots,m_{\ell})\in\oplus_{i=1}^{\ell}\widetilde{E}(M_{i})\subseteq\oplus_{k\in\Lambda}\widetilde{E}(M_{k})$.
Therefore $\widetilde{E}(M)\leq\oplus_{k\in\Lambda}\widetilde{E}(M_{k})$. ∎

######  Remark 3.3.

Example 3.1 illustrates Proposition 3.2 because $R=\mathbb{Z}$ is a noetherian
ring, that is,
$\widetilde{E}(\mathbb{Z}\oplus\mathbb{Z}_{p})=\mathbb{Z}_{(p)}\oplus\mathbb{Z}_{p}\lneq\mathbb{Q}\oplus\mathbb{Z}_{p}=\widetilde{E}(\mathbb{Z})\oplus\widetilde{E}(\mathbb{Z}_{p})$.
However, Example 3.7 shows that the condition “either $R$ is right noetherian
or $|\Lambda|$ is finite” is not superfluous because
$\widetilde{E}(\oplus_{k\in\Lambda}\mathbb{Z}_{2})=\prod_{k\in\Lambda}\mathbb{Z}_{2}\gneq\oplus_{k\in\Lambda}\mathbb{Z}_{2}=\oplus_{k\in\Lambda}\widetilde{E}(\mathbb{Z}_{2})$
with a non-noetherian ring
$R=\langle\oplus_{k\in\Lambda}\mathbb{Z}_{2},1\rangle$.

To get the reverse inclusion of Proposition 3.2, first we provide the
properties of the relatively dense property.

######  Lemma 3.4.

Let $N\leq M$ and $K_{i}$ be right $R$-modules for all $i\in\Lambda$. Then $N$
is $K_{i}$-dense in $M$ for all $i\in\Lambda$ if and only if $N$ is
$\bigoplus_{i\in\Lambda}K_{i}$-dense in $M$ if and only if $N$ is
$\bigoplus_{i\in\Lambda}\widetilde{E}(K_{i})$-dense in $M$.

###### Proof.

Let $P$ be any submodule such that $N\leq P\leq M$. Since $N$ is $K_{i}$-dense
in $M$, $\text{Hom}_{R}(P/N,K_{i})=0$ for all $i\in\Lambda$ from Proposition
2.5. Consider the sequence
$0\rightarrow\oplus_{i\in\Lambda}K_{i}\rightarrow\prod_{i\in\Lambda}K_{i}$.
Then we have
$0\rightarrow\text{Hom}_{R}(P/N,\oplus_{i\in\Lambda}K_{i})\rightarrow\text{Hom}_{R}(P/N,\prod_{i\in\Lambda}K_{i})\cong\prod_{i\in\Lambda}(\text{Hom}_{R}(P/N,K_{i})=0$.
Thus $\text{Hom}_{R}(P/N,\oplus_{i\in\Lambda}K_{i})=0$. Therefore $N$ is
$\oplus_{i\in\Lambda}K_{i}$-dense in $M$ from Proposition 2.5. Conversely,
since $\text{Hom}_{R}(P/N,\oplus_{i\in\Lambda}K_{i})=0$,
$\text{Hom}_{R}(P/N,K_{i})=0$ for each $i\in\Lambda$. Hence $N$ is
$K_{i}$-dense in $M$ for all $i\in\Lambda$.
For the second equivalence, since $E(\widetilde{E}(K_{i}))=E(K_{i})$, from
Proposition 2.5 it is easy to see that $N$ is $K_{i}$-dense in $M$ if and only
if $N$ is $\widetilde{E}(K_{i})$-dense in $M$, for all $i\in\Lambda$. After
using the first equivalence, we have the second equivalence. ∎

Using Lemma 3.4, we obtain a characterization for $\oplus_{k\in\Lambda}N_{k}$
to be a dense submodule of $\oplus_{k\in\Lambda}M_{k}$ where $N_{i}$ is a
submodule of $M_{i}$ for each $i\in\Lambda$.

######  Proposition 3.5.

Let $N_{i}\leq M_{i}$ be right $R$-modules for all $i\in\Lambda$ where
$\Lambda$ is any index set. Let $N=\bigoplus_{k\in\Lambda}N_{k}$ and
$M=\bigoplus_{k\in\Lambda}M_{k}$. Then $N\leq^{\emph{den}}M$ if and only if
$N_{i}$ is $M_{j}$-dense in $M_{i}$ for all $i,j\in\Lambda$.

###### Proof.

Suppose $N\leq^{\text{den}}M$. Then $N$ is $M$-dense in $M$ by the definition.
From Lemma 3.4 $N$ is $M_{j}$-dense in $M$ for all $j\in\Lambda$. Let
$x_{i}\in M_{i}$ and $0\neq y_{j}\in M_{j}$ be arbitrary for each
$i,j\in\Lambda$. Since $(0,\dots,0,x_{i},0,\dots)\in M$ and $0\neq y_{j}\in
M_{j}$, there exists $r\in R$ such that $(0,\dots,0,x_{i},0,\dots)r\in N$ and
$y_{j}r\neq 0$. Since $x_{i}r\in N_{i}$ and $y_{j}r\neq 0$, $N_{i}$ is
$M_{j}$-dense in $M_{i}$ for all $i,j\in\Lambda$.

Conversely, suppose $N_{i}$ is $M_{j}$-dense in $M_{i}$ for all
$i,j\in\Lambda$. From Lemma 3.4, $N_{i}$ is $\oplus_{k\in\Lambda}M_{k}$-dense
in $M_{i}$ for all $i\in\Lambda$. Let $x\in M$ and $0\neq y\in M$ be
arbitrary. Then there exists $\ell\in\mathbb{N}$ such that
$x=(x_{1},\dots,x_{\ell})\in\oplus_{k=1}^{\ell}M_{k}\leq M$. Since $N_{1}$ is
$M$-dense in $M_{1}$, there exists $r_{1}\in R$ such that $x_{1}r_{1}\in
N_{1}$ and $0\neq yr_{1}\in M$. Also, since $N_{2}$ is $M$-dense in $M_{2}$,
there exists $r_{2}\in R$ such that $x_{2}r_{1}r_{2}\in N_{2}$ and $0\neq
yr_{1}r_{2}\in M$. By the similar processing, we have $r=r_{1}r_{2}\cdots
r_{\ell}\in R$ such that $xr\in\oplus_{k=1}^{\ell}N_{k}\leq N$ and $yr\neq 0$.
Therefore $N\leq^{\text{den}}M$. ∎

From Propositions 3.2 and 3.5, we have a characterization for the rational
hull of the direct sum of modules to be the direct sum of the rational hulls
of each module.

######  Theorem 3.6.

Let $M=\bigoplus_{k\in\Lambda}M_{k}$ where $M_{k}$ is a right $R$-module and
$\Lambda$ is any index set. If either $R$ is right noetherian or $|\Lambda|$
is finite, then $\widetilde{E}(M)=\bigoplus_{k\in\Lambda}\widetilde{E}(M_{k})$
if and only if $M_{i}$ is $M_{j}$-dense in $\widetilde{E}(M_{i})$ for all
$i,j\in\Lambda$.

###### Proof.

Suppose $\widetilde{E}(M)=\oplus_{k\in\Lambda}\widetilde{E}(M_{k})$. Since
$M\leq^{\text{den}}\oplus_{k\in\Lambda}\widetilde{E}(M_{k})$, from Proposition
3.5 $M_{i}$ is $\widetilde{E}(M_{j})$-dense in $\widetilde{E}(M_{i})$ for all
$i,j\in\Lambda$. Thus, $M_{i}$ is $M_{j}$-dense in $\widetilde{E}(M_{i})$ for
all $i,j\in\Lambda$ from Lemma 3.4.

Conversely, suppose $M_{i}$ is $M_{j}$-dense in $\widetilde{E}(M_{i})$ for all
$i,j\in\Lambda$. Then $M_{i}$ is $\widetilde{E}(M_{j})$-dense in
$\widetilde{E}(M_{i})$ for all $i,j\in\Lambda$ from Lemma 3.4. Thus, from
Proposition 3.5 $M\leq^{\text{den}}\oplus_{k\in\Lambda}\widetilde{E}(M_{k})$.
Hence $\oplus_{k\in\Lambda}\widetilde{E}(M_{k})\leq\widetilde{E}(M)$ from
Proposition 1.3. Also, from Proposition 3.2
$\widetilde{E}(M)\leq\oplus_{k\in\Lambda}\widetilde{E}(M_{k})$. Therefore
$\widetilde{E}(M)=\oplus_{k\in\Lambda}\widetilde{E}(M_{k})$. ∎

The next examples show that the condition “$R$ is right noetherian or
$|\Lambda|$ is finite” in Theorem 3.6 is not superfluous.

######  Example 3.7.

(i) Let $R=\langle\oplus_{k\in\Lambda}\mathbb{Z}_{2},1\rangle$ and
$M=\oplus_{k\in\Lambda}M_{k}$ where $M_{k}=\mathbb{Z}_{2}$. Note that $R$ is
not noetherian. Since $\mathbb{Z}_{2}$ is an injective $R$-module,
$\widetilde{E}(\mathbb{Z}_{2})=\mathbb{Z}_{2}$. Thus $M_{i}$ is $M_{j}$-dense
in $\widetilde{E}(M_{i})$ for all $i,j\in\Lambda$. However,
$\widetilde{E}(\oplus_{k\in\Lambda}\mathbb{Z}_{2})=\prod_{k\in\Lambda}\mathbb{Z}_{2}\gneq\oplus_{k\in\Lambda}\mathbb{Z}_{2}=\oplus_{k\in\Lambda}\widetilde{E}(\mathbb{Z}_{2})$.

(ii) Let $R=\\{(a_{k})\in\prod_{k\in\Lambda}\mathbb{Z}\,|\,a_{k}~{}\text{is
eventually constant}\\}$ and $M=\oplus_{k\in\Lambda}\mathbb{Z}$. Note that $R$
is not noetherian. Then $\widetilde{E}(\mathbb{Z})=\mathbb{Q}$ and
$\mathbb{Z}$ is $\mathbb{Z}$-dense in $\widetilde{E}(\mathbb{Z})$. However,
$\widetilde{E}(\oplus_{k\in\Lambda}\mathbb{Z})=\prod_{k\in\Lambda}\mathbb{Q}\gneq\oplus_{k\in\Lambda}\mathbb{Q}=\oplus_{k\in\Lambda}\widetilde{E}(\mathbb{Z})$.

The next example illustrates Theorem 3.6.

######  Example 3.8.

Consider $M=\mathbb{Z}\oplus\mathbb{Z}_{p}$ as a $\mathbb{Z}$-module where $p$
is prime. Then $\mathbb{Z}_{p}$ is $\mathbb{Z}$-dense in
$\widetilde{E}(\mathbb{Z}_{p})=\mathbb{Z}_{p}$, but $\mathbb{Z}$ is not
$\mathbb{Z}_{p}$-dense in $\mathbb{Q}$ because for
$\frac{1}{p}\in\mathbb{Q},\overline{1}\in\mathbb{Z}_{p}$, there is no element
$t\in\mathbb{Z}$ such that $t\frac{1}{p}\in\mathbb{Z}$ and $t\overline{1}\neq
0$. Thus, from Theorem 3.6
$\widetilde{E}(M)=\mathbb{Z}_{(p)}\oplus\mathbb{Z}_{p}\lneq\mathbb{Q}\oplus\mathbb{Z}_{p}=\widetilde{E}(\mathbb{Z})\oplus\widetilde{E}(\mathbb{Z}_{p})$.
(See Example 3.1 for details.)

######  Corollary 3.9.

Let $M$ be a right $R$-module. If either $R$ is right noetherian or $\Lambda$
is a finite index set, then
$\widetilde{E}(M^{(\Lambda)})=(\widetilde{E}(M))^{(\Lambda)}$.

######  Corollary 3.10.

Let $\\{M_{k}\\}_{k\in\Lambda}$ be a class of rationally complete right
$R$-module for any index set $\Lambda$. If either $R$ is right noetherian or
$|\Lambda|$ is finite, then $M=\bigoplus_{k\in\Lambda}M_{k}$ is rationally
complete.

###### Proof.

Since $\widetilde{E}(M_{i})=M_{i}$, $M_{i}$ is $M_{j}$-dense in
$\widetilde{E}(M_{i})$ for all $i,j\in\Lambda$. From Theorem 3.6,
$\widetilde{E}(M)=\oplus_{k\in\Lambda}\widetilde{E}(M_{k})=\oplus_{k\in\Lambda}M_{k}=M$.
Therefore $\oplus_{k\in\Lambda}M_{k}$ is rationally complete. ∎

######  Proposition 3.11 ([14, Proposition 1.9]).

Let $\\{S_{i}\\}_{i\in\Lambda}$ be a set of nonisomorphic simple modules,
representing all singular simple modules. Then every module containing the
module $P=\bigoplus_{i\in\Lambda}S_{i}$ is rationally complete.

##  4\. The endomorphism ring of a module over a right ring of quotients of a
ring

In this section, we obtain some condition to be
$\text{End}_{R}(M)=\text{End}_{H}(M)$ where $H$ is a right ring of quotients
of a ring $R$. Recall that an extension ring $H$ of a ring $R$ is called a
_right ring of quotients_ of $R$ if for any two elements $x\neq 0$ and $y$ of
$H$, there exists an element $r\in R$ such that $xr\neq 0$ and $yr\in R$.

######  Theorem 4.1.

Let $M$ be a right $H$-module where $H$ is a right ring of quotients of a ring
$R$. If $R$ is $M_{R}$-dense in $H_{R}$ then
$\emph{End}_{R}(M)=\emph{End}_{H}(M)$.

###### Proof.

Since $\text{End}_{H}(M)\subseteq\text{End}_{R}(M)$, it suffices to show that
$\text{End}_{R}(M)\subseteq\text{End}_{H}(M):$ Let
$\varphi\in\text{End}_{R}(M)$ be arbitrary. Assume that
$\varphi\notin\text{End}_{H}(M)$. Then there exist $m\in M,t\in H$ such that
$\varphi(mt)-\varphi(m)t\neq 0.$ Since $R$ is $M_{R}$-dense in $H_{R}$, there
exists $r\in R$ such that $(\varphi(mt)-\varphi(m)t)r\neq 0$ and $tr\in R$.
Hence
$0\neq(\varphi(mt)-\varphi(m)t)r=\varphi(mt)r-\varphi(m)(tr)=\varphi(mtr)-\varphi(mtr)=0$,
a contradiction. Therefore $\text{End}_{R}(M)=\text{End}_{H}(M)$. ∎

######  Remark 4.2.

(i) $R$ is always $E(R)$-dense in $H_{R}$ where $H$ is a right ring of
quotients of $R$. For, let $x\in H_{R}$ and $0\neq y\in E(R)$. Since
$H\leq^{\text{ess}}E(R)_{R}$, there exists $s\in R$ such that $0\neq ys\in H$.
Also, $xs\in H$. Since $R\leq^{\text{den}}H_{R}$, there exists $t\in R$ such
that $xst\in R$ and $0\neq yst$. Therefore $R$ is $E(R)$-dense in $H_{R}$.
(ii) If $M$ is a nonsingular $R$-module, then $R$ is $M_{R}$-dense in $H_{R}$:
For, let $0\neq m\in M$ and $t\in H$ be arbitrary. Take $t^{-1}R=\\{r\in
R\,|\,tr\in R\\}$ a right ideal of $R$. Note that
$t^{-1}R\leq^{\textrm{ess}}R_{R}$. Since
$t^{-1}R\nleq^{\text{ess}}\mathbf{r}_{R}(m)$, there exists $r\in t^{-1}R$ and
$r\notin\mathbf{r}_{R}(m)$. Thus, $tr\in R$ and $mr\neq 0$. Therefore $R$ is
$M_{R}$-dense in $H_{R}$.
(iii) If $M$ is a submodule of a projective right $H$-module, then $R$ is
$M_{R}$-dense in $H_{R}$. For, let $P$ be a projective right $R$-module
including $M$, that is, $M\leq P$ where $P\leq^{\oplus}H^{(\Lambda)}$ with
some index set $\Lambda$. Then there is a right $R$-module $K\leq E(P)$ such
that $E(P)=E(M)\oplus K.$ Since $R\leq^{\text{den}}H_{R}$, we get that $R$ is
$H^{(\Lambda)}$-dense in $H_{R}$ from Lemma 3.4. Hence $R$ is $P$-dense in
$H_{R}$. Thus $\text{Hom}_{R}(H/R,E(P))=0$ from Proposition 2.5. Since
$\text{Hom}_{R}(H/R,E(P))\cong\text{Hom}_{R}(H/R,E(M))\oplus\text{Hom}_{R}(H/R,K)$,
we obtain $\text{Hom}_{R}(H/R,E(M))=0$. It follows that $R$ is $M_{R}$-dense
in $H_{R}.$

######  Corollary 4.3.

Let $M$ be a projective right $H$-module where $H$ is a right ring of
quotients of $R$. Then $\emph{End}_{R}(M)=\emph{End}_{H}(M)$.

The next example illustrates Corollary 4.3.

######  Example 4.4.

Let $Q=\prod_{n=1}^{\infty}\mathbb{Z}_{2}$ and $R=\\{(a_{n})\in Q~{}|~{}a_{n}$
is eventually constant$\\}$. Then $Q$ is a maximal right ring of quotients of
$R$. Hence from Theorem 4.1, End${}_{R}(Q^{(\Lambda)})$=
End${}_{Q}(Q^{(\Lambda)})=\mathsf{CFM}_{\Lambda}(Q)$.

######  Theorem 4.5.

Let $M$ be a finitely generated free $R$-module with $S=\emph{End}_{R}(M)$. If
either $R$ is right noetherian or $\Lambda$ is any finite index set, then
$\emph{End}_{R}\left(\widetilde{E}(M^{(\Lambda)})\right)=\mathsf{CFM}_{\Lambda}\left(Q(S)\right)$.

###### Proof.

Let $M=R^{(n)}$ for some $n\in\mathbb{N}$. From Corollary 3.9,
$\widetilde{E}(R^{(n)})=\widetilde{E}(R)^{(n)}=Q(R)^{(n)}$ as
$\widetilde{E}(R)=Q(R)$. Hence
$\text{End}_{R}\left(\widetilde{E}(M^{(\Lambda)})\right)=\text{End}_{R}\left(\widetilde{E}(M)^{(\Lambda)}\right)=\text{End}_{R}\left((Q(R)^{(n)})^{(\Lambda)}\right)=\text{End}_{Q(R)}\left((Q(R)^{(n)})^{(\Lambda)}\right)=\mathsf{CFM}_{\Lambda}\left(\text{End}_{Q(R)}(Q(R)^{(n)})\right)=\mathsf{CFM}_{\Lambda}\left(\mathsf{Mat}_{n}(Q(R))\right)$
from Theorem 4.1. Therefore
$\text{End}_{R}\left(\widetilde{E}(M^{(\Lambda)})\right)=\mathsf{CFM}_{\Lambda}\left(Q(\text{End}_{R}(M))\right)$
because $\mathsf{Mat}_{n}(Q(R))=Q(\mathsf{Mat}_{n}(R))$ by [15, 2.3] and
$\text{End}_{R}(M)=\mathsf{Mat}_{n}(R)$. ∎

The next result is generalized from [15, 2.3].

######  Corollary 4.6.

Let $M$ be a finitely generated free $R$-module. Then
$Q(\emph{End}_{R}(M))=\emph{End}_{R}(\widetilde{E}(M))$.

The following example shows that the above result can not be extended to flat
modules. This example also shows that $R$ is not $M_{R}$-dense in $Q_{R}$
where $Q$ is a right ring of quotients of $R$.

######  Example 4.7.

Let $Q=\prod_{n=1}^{\infty}\mathbb{Z}_{2}$, $R=\\{(a_{n})\in Q~{}|~{}a_{n}$ is
eventually constant$\\}$, and $I=\\{(a_{n})\in
Q~{}|~{}a_{n}=0~{}\textrm{eventually}\\}$. Note that $Q=Q(R)$. Let $M=Q/I$,
which is a flat $Q$-module but not projective. We claim that
End${}_{Q}(M)\subsetneq$ End${}_{R}(M)$. Indeed, define $f:M\to M$ via

$f[(a_{1},a_{2},\dots,a_{n},a_{n+1},\dots)+I]=(a_{1},0,a_{2},0,\dots,a_{n},0,a_{n+1},0,\dots)+I,$

for any $\overline{a}=a+I=(a_{1},a_{2},\dots,a_{n},a_{n+1},\dots)+I\in M$. It
is easy to see that $f(\overline{a}+\overline{b})$ = $f(\overline{a})$ \+
$f(\overline{b})$ for any $\overline{a}$, $\overline{b}\in M$. Meanwhile, for
any $r=(r_{1},r_{2},\dots,r_{n},r_{n+1},\dots)\in R$, we have

$(a+I)r=ar+I=\left\\{\begin{array}[]{ll}(0,\dots,0,a_{n},a_{n+1},\dots)+I,&\mbox{if
}r_{n}=r_{n+1}=\dots=1;\\\ (0,\dots,0,0,0,\dots)+I,&\mbox{if
}r_{n}=r_{n+1}=\dots=0.\end{array}\right.$

Note that $a+I=(0,0,\dots,0,a_{n},a_{n+1},\dots)+I$ for some $n\in\mathbb{N}$.
One can easily see that $f[(a+I)r]$ = $[f(a+I)]r$ for all $a\in Q$, $r\in R$.
This shows $f\in$ End${}_{R}(M)$. However, for $q=(0,1,0,1,\dots)=q^{2}\in Q$,
we have $[f(q+I)]q=0+I$ while $f[(q+I)q]=f(q+I)\neq 0+I$. This means
$f\not\in$ End${}_{Q}(M)$. Thus, End${}_{Q}(M)\subsetneq$ End${}_{R}(M)$. Note
that $R$ is not $M_{R}$-dense in $Q$. For, let $q\in Q\setminus R$ and
$m=1+I\in M$. Since $(1+I)r=0+I$ for all $r\in R\setminus{1}$, it has to be
$r=1$ to get $mr\neq 0+I$. However, $qr\notin R$.

Recall that a module $M$ is said to be _polyform_ if every essential submodule
of $M$ is a dense submodule.

######  Lemma 4.8.

A module $M$ is polyform iff $\widetilde{E}(M)$ is a polyform quasi-injective
module.

###### Proof.

Let $X$ be essential in $\widetilde{E}(M)$. Then $X\cap M\leq^{\text{ess}}M$.
Hence $X\cap M$ is a dense submodule of $M$ because $M$ is polyform. Since
$X\cap M\leq^{\text{den}}M\leq^{\text{den}}\widetilde{E}(M)$, $X\cap
M\leq^{\text{den}}\widetilde{E}(M)$. Thus $X$ is a dense submodule of
$\widetilde{E}(M)$ from Proposition 1.2(ii). Therefore $\widetilde{E}(M)$ is a
polyform module. In addition, $\widehat{M}$ is also a polyform module from
[16, 11.1]. Since $M\leq^{\text{ess}}\widehat{M}$,
$M\leq^{\text{den}}\widehat{M}$. Thus
$\widetilde{E}(M)=\widetilde{E}(\widehat{M})$. Since the rational hull of a
quasi-injective module is also quasi-injective from Theorem 2.16,
$\widetilde{E}(M)$ is a quasi-injective module. Therefore $\widetilde{E}(M)$
is a polyform quasi-injective module.

Conversely, let $N$ be any essential submodule of $M$. Then $N$ is also
essential in $\widetilde{E}(M)$. Hence $N$ is a dense submodule of
$\widetilde{E}(M)$ as $\widetilde{E}(M)$ is polyform. So $N$ is a dense
submodule of $M$. Therefore $M$ is polyform. ∎

We show from Theorem 2.15 that there is a canonical embedding of the ring
$\text{End}_{R}(M)$ into the ring $\text{End}_{R}(\widetilde{E}(M))$. Next, we
obtain a condition when $\text{End}_{R}(M)$ and
$\text{End}_{R}(\widetilde{E}(M))$ are isomorphic. It is a generalization of
[7, Exercises 7.32].

######  Proposition 4.9.

If $M$ is a quasi-injective module then
$\emph{End}_{R}(M)\stackrel{{\scriptstyle\Omega}}{{\cong}}\emph{End}_{R}(\widetilde{E}(M))$.
In particular, if $M$ is a polyform module, then the converse holds true.

###### Proof.

In the proof of Theorem 2.15, we only need to show that
$\Omega:\text{End}_{R}(M)\rightarrow\text{End}_{R}(\widetilde{E}(M))$ given by
$\Omega(\varphi)=\widetilde{\varphi}$, is surjective: Let
$\psi\in\text{End}_{R}(\widetilde{E}(M))$ be arbitrary. Then there exists
$\widehat{\psi}\in\text{End}_{R}(E(M))$ such that
$\widehat{\psi}|_{\widetilde{E}(M)}=\psi$. Since $\widehat{\psi}M\leq M$ as
$M$ is quasi-injective, $\widehat{\psi}|_{M}=\psi|_{M}\in\text{End}_{R}(M)$.
Thus, $\Omega(\psi|_{M})=\psi$, which shows that $\Omega$ is surjective.

In addition, suppose that $M$ is a polyform module. Then from Lemma 4.8,
$\widetilde{E}(M)$ is quasi-injective. Thus, for any
$\vartheta\in\text{End}_{R}(E(M))$,
$\vartheta\widetilde{E}(M)\leq\widetilde{E}(M)$. Since
$\vartheta|_{\widetilde{E}(M)}\in\text{End}_{R}(\widetilde{E}(M))$ and
$\text{End}_{R}(M)\stackrel{{\scriptstyle\Omega}}{{\cong}}\text{End}_{R}(\widetilde{E}(M))$,
there exists $\varphi\in\text{End}_{R}(M)$ such that
$\Omega(\varphi)=\vartheta|_{\widetilde{E}(M)}$. Also by Theorem 2.15,
$\vartheta|_{M}=\varphi$. Thus, $\vartheta M=\varphi M\leq M$. Therefore $M$
is a quasi-injective module. ∎

######  Corollary 4.10.

If $M$ is a quasi-injective module, then
$Q(\emph{End}_{R}(M))\cong\emph{End}_{R}(\widetilde{E}(M))$.

###### Proof.

Since $M$ is a quasi-injective module $\text{End}_{R}(M)$ is a right self-
injective ring. So, $Q(\text{End}_{R}(M))=\text{End}_{R}(M)$. Thus,
$Q(\text{End}_{R}(M))\cong\text{End}_{R}(\widetilde{E}(M))$ by Proposition
4.9. ∎

Remark that if $M$ is a quasi-injective module then $\widetilde{E}(M)$ is a
quasi-injective module from Theorem 2.16 and
$\text{End}_{R}(M)\cong\text{End}_{R}(\widetilde{E}(M))$ from Proposition 4.9.
However, the next example shows that there exists a quasi-injective module $M$
such that $M\neq\widetilde{E}(M)$.

######  Example 4.11.

Let $R=\left(\begin{smallmatrix}F&F\\\ 0&F\end{smallmatrix}\right)$ and
$M=\left(\begin{smallmatrix}0&0\\\ 0&F\end{smallmatrix}\right)$ where $F$ is a
field. Then $M$ is a quasi-injective $R$-module. However,
$\widetilde{E}(M)=E(M)=\left(\begin{smallmatrix}0&0\\\
F&F\end{smallmatrix}\right)$ because $M$ is nonsingular. Thus $M$ is a quasi-
injective $R$-module such that $M\lneq\widetilde{E}(M)$ and
$\text{End}_{R}(M)\cong\left(\begin{smallmatrix}0&0\\\
0&F\end{smallmatrix}\right)\cong\text{End}_{R}(\widetilde{E}(M))$.

Because $\widetilde{E}(M)=E(M)$ for a right nonsingular module $M$, we have
the following well-known results as a consequence of Proposition 4.9.

######  Corollary 4.12 ([7, Exercises 7.32]).

For any nonsingular module $M$, the following statements hold true:

  1. (i)

there is a canonical embedding $\Omega$ of the ring $\emph{End}_{R}(M)$ into
the ring $\emph{End}_{R}(E(M))$.

  2. (ii)

$M$ is a quasi-injective $R$-module if and only if $\Omega$ is an isomorphism.

######  Corollary 4.13.

Let $M$ be a right $H$-module where $H$ is a right ring of quotients of a ring
$R$. The following statements hold true:

  1. (i)

If $M$ is a nonsingular $R$-module then $\emph{End}_{R}(M)=\emph{End}_{H}(M)$.

  2. (ii)

If $M$ is a submodule of a projective $H$-module, then
$\emph{End}_{R}(M)=\emph{End}_{H}(M)$.

  3. (iii)

If $M$ is a nonsingular quasi-injective $R$-module then
$\emph{End}_{R}(M)\cong\emph{End}_{R}(E(M))$ and
$\emph{End}_{H}(M)\cong\emph{End}_{H}(E(M))$.

  4. (iv)

If $M$ is a quasi-injective $R$-module and is a submodule of a projective
$H$-module then $\emph{End}_{R}(M)\cong\emph{End}_{R}(\widetilde{E}(M))$ and
$\emph{End}_{H}(M)\cong\emph{End}_{H}(\widetilde{E}(M))$.

## References

  * [1] E. Akalan; G.F. Birkenmeier; A. Tercan, Goldie extending modules, Comm. Algebra, 2009 _37(2)_ , 663–683
  * [2] K. Asano, Über die Quotientenbildung von Schiefringen, J. Math. Soc. Japan, 1949 _1(2)_ , 73–78
  * [3] G.F. Birkenmeier; J.K. Park; S.T. Rizvi, _Extensions of Rings and Modules_ , Birkhäuser/Springer, New York-Heidelberg-Dordrecht-London (2013)
  * [4] G.D. Findlay; J. Lambek, A generalized ring of quotients I, Canad. Math. Bull., 1958 _1_ , 77–85
  * [5] A.W. Goldie, Localization in non-commutative noetherian rings, J. Algebra, 1967 _5_ , 89–105
  * [6] R.E. Johnson, The extended centralizer of a ring over a module, Proc. Amer. Math. Soc., 1951 _2_ , 891–895
  * [7] T.Y. Lam, _Lectures on Modules and Rings_ , GTM 189, Springer Verlag, Berlin-Heidelberg-New York (1999)
  * [8] J. Lambek, On Utumi’s ring of quotients, Canadian J. Math., 1963 _15_ , 363–370
  * [9] J. Lambek, _Lectures on Rings and Modules_ , Blaisdell Publishing Co. [Ginn and Co.], Waltham, Mass.-Toronto, Ont.-London (1966)
  * [10] S.H. Mohamed; B.J. M$\ddot{\textrm{u}}$ller, _Continuous and Discrete Modules_ , Lecture Note Ser. No. 147, London Math. Soc., Cambridge Univ. Press (1990)
  * [11] Ø. Ore, Linear equations in non-commutative fields, Ann. of Math. Second Series, 1931 _32(3)_ , 463–477
  * [12] S.T. Rizvi; C.S. Roman, On $\mathcal{K}$-nonsingular modules and applications, Comm. Algebra, 2007 _35_ , 2960–2982
  * [13] B. Stenström, _Rings of Quotients_ , Band 217, Springer Verlag, Berlin-Heidelberg-New York (1975)
  * [14] H. Storrer, Rational extensions of modules, Pacific Journal of Mathematics, 1971 _38(3)_ , 785–794
  * [15] Y. Utumi, On quotient rings, Osaka Math. J., 1956 _8_ , 1–18
  * [16] R. Wisbauer, _Modules and Algebras: bimodule structure and group actions on algebras_ , Pitman Monographs and Surveys in Pure and Applied Mathematics 81, Longman, Harlow (1996)
  * [17] X. Zhang; G. Lee; N.K. Tung, Relatively polyform modules, Journal of Algebra Appl. 2025 2550046 (17 pages) DOI: 10.1142/S021949882550046X