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....

Module is said to be small if it is not a union of strictly increasing infinite countable chain of submodules. We show that the class of all small modules over self-injective purely infinite ring is closed under direct products whenever there exists no strongly inaccessible cardinal.

The structure theory of abelian $p$-groups does not depend on the properties of the ring of integers, in general. The substantial portion of this theory is based on the fact that a finitely generated $p$-group is a direct sum of cyclics. Given a hereditary torsion theory on the category $R$-Mod of unitary left $R$-modules we can investigate torsionfree modules having the corresponding property for all torsionfree factor-modules (and a natural requirement concerning extensions of some homomorphisms). This...

Rim and Teply [10] investigated relatively exact modules in connection with the existence of torsionfree covers. In this note we shall study some properties of the lattice ${ℰ}_{\tau }\left(M\right)$ of submodules of a torsionfree module $M$ consisting of all submodules $N$ of $M$ such that $M/N$ is torsionfree and such that every torsionfree homomorphic image of the relative injective hull of $M/N$ is relatively injective. The results obtained are applied to the study of relatively exact covers of torsionfree modules. As an application...

We investigate the representation theory of the positively based algebra $A_{m,d}$, which is a generalization of the noncommutative Green algebra of weak Hopf algebra corresponding to the generalized Taft algebra. It turns out that $A_{m,d}$ is of finite representative type if $d\le 4$, of tame type if $d=5$, and of wild type if $d\ge 6.$ In the case when $d\le 4$, all indecomposable representations of $A_{m,d}$ are constructed. Furthermore, their right cell representations as well as left cell representations of $A_{m,d}$ are described.

A right $R$-module $M$ is called $R$-projective provided that it is projective relative to the right $R$-module $R_R$. This paper deals with the rings whose all nonsingular right modules are $R$-projective. For a right nonsingular ring $R$, we prove that $R_R$ is of finite Goldie rank and all nonsingular right $R$-modules are $R$-projective if and only if $R$ is right finitely $\Sigma$-$CS$ and flat right $R$-modules are $R$-projective. Then, $R$-projectivity of the class of nonsingular injective right modules is also considered. Over right...

Let M be a left module over a ring R. M is called a Zelmanowitz-regular module if for each x ∈ M there exists a homomorphism F: M → R such that f(x) = x. Let Q be a left R-module and h: Q → M a homomorphism. We call h locally split if for every x ∈ M there exists a homomorphism g: M → Q such that h(g(x)) = x. M is called locally projective if every epimorphism onto M is locally split. We prove that the following conditions are equivalent:(1) M is Zelmanowitz-regular.(2) every homomorphism into M...