We study the class of modules which are invariant under idempotents of their envelopes. We say that a module M is -idempotent-invariant if there exists an -envelope u : M → X such that for any idempotent g ∈ End(X) there exists an endomorphism f : M → M such that uf = gu. The properties of this class of modules are discussed. We prove that M is -idempotent-invariant if and only if for every decomposition $X={⨁}_{i\in I}{X}_i$, we have $M={⨁}_{i\in I}\left(u^{-1}\left(X_i\right)\cap M\right)$. Moreover, some generalizations of -idempotent-invariant modules are considered....

We characterize the semiregularity of the endomorphism ring of a module with respect to the ideal of endomorphisms with large kernel, and show some new classes of modules with semiregular endomorphism rings.

We shall introduce the class of strongly cancellative multiplicative monoids which contains the class of all totally ordered cancellative monoids and it is contained in the class of all cancellative monoids. If $G$ is a strongly cancellative monoid such that $hG\subseteq Gh$ for each $h\in G$ and if $R$ is a ring such that $aR\subseteq Ra$ for each $a\in R$, then the class of all non-singular left $R$-modules is a cover class if and only if the class of all non-singular left $RG$-modules is a cover class. These two conditions are also equivalent whenever...

Let $G$ be a multiplicative monoid. If $RG$ is a non-singular ring such that the class of all non-singular $RG$-modules is a cover class, then the class of all non-singular $R$-modules is a cover class. These two conditions are equivalent whenever $G$ is a well-ordered cancellative monoid such that for all elements $g,h\in G$ with $g<h$ there is $l\in G$ such that $lg=h$. For a totally ordered cancellative monoid the equalities $Z\left(RG\right)=Z\left(R\right)G$ and $\sigma \left(RG\right)=\sigma \left(R\right)G$ hold, $\sigma$ being Goldie’s torsion theory.

One of the results in my previous paper On torsionfree classes which are not precover classes, preprint, Corollary 3, states that for every hereditary torsion theory $\tau$ for the category $R$-mod with $\tau \ge \sigma$, $\sigma$ being Goldie’s torsion theory, the class of all $\tau$-torsionfree modules forms a (pre)cover class if and only if $\tau$ is of finite type. The purpose of this note is to show that all members of the countable set $𝔐=\{R,R/\sigma \left(R\right),R[x_1,\cdots ,x_n],R[x_1,\cdots ,x_n]/\sigma \left(R[x_1,\cdots ,x_n]\right),n<\omega \}$ of rings have the property that the class of all non-singular left modules forms a (pre)cover...

Let X be a class or R-modules containing 0 and closed under isomorphic images. With any such X we associate three classes ΓX, FX and ΔX. The study of some of the closure properties of these classes allows us to obtain characterization of Artinian modules dualizing results of Chatters. The theory of Dual Glodie dimension as developed by the author in some of his earlier work plays a crucial role in the present paper.

We introduce the notion of FI-mono-retractable modules which is a generalization of compressible modules. We investigate the properties of such modules. It is shown that the rings over which every cyclic module is FI-mono-retractable are simple Noetherian $V$-ring with zero socle or Artinian semisimple. The last section of the paper is devoted to the endomorphism rings of FI-retractable modules.

A module M satisfies the restricted minimum condition if M/N is artinian for every essential submodule N of M. A ring R is called a right RM-ring whenever $R_R$ satisfies the restricted minimum condition as a right module. We give several structural necessary conditions for particular classes of RM-rings. Furthermore, a commutative ring R is proved to be an RM-ring if and only if R/Soc(R) is noetherian and every singular module is semiartinian.

Let $R$ be a ring. A right $R$-module $M$ is said to be retractable if $𝕋{Hom}_R(M,N)\ne 0$ whenever $N$ is a non-zero submodule of $M$. The goal of this article is to investigate a ring $R$ for which every right R-module is retractable. Such a ring will be called right mod-retractable. We proved that $\left(1\right)$ The ring ${\prod }_{i\in ℐ}{R}_i$ is right mod-retractable if and only if each $R_i$ is a right mod-retractable ring for each $i\in ℐ$, where $ℐ$ is an arbitrary finite set. $\left(2\right)$ If $R\left[x\right]$ is a mod-retractable ring then $R$ is a mod-retractable ring.

The aim of this paper is to investigate quasi-corational, comonoform, copolyform and $\alpha$-(co)atomic modules. It is proved that for an ordinal $\alpha$ a right $R$-module $M$ is $\alpha$-atomic if and only if it is $\alpha$-coatomic. And it is also shown that an $\alpha$-atomic module $M$ is quasi-projective if and only if $M$ is quasi-corationally complete. Some other results are developed.

In this paper we introduce the concept of $\tau$-extending modules by $\tau$-rational submodules and study some properties of such modules. It is shown that the set of all $\tau$-rational left ideals of ${}_{R}R$ is a Gabriel filter. An $R$-module $M$ is called $\tau$-extending if every submodule of $M$ is $\tau$-rational in a direct summand of $M$. It is proved that $M$ is $\tau$-extending if and only if $M=Re{j}_{M}E(R/\tau \left(R\right))\oplus N$, such that $N$ is a $\tau$-extending submodule of $M$. An example is given to show that the direct sum of $\tau$-extending modules need not be $\tau$-extending....