We give some new characterizations of quasi-Frobenius rings by the generalized injectivity of rings. Some characterizations give affirmative answers to some open questions about quasi-Frobenius rings; and some characterizations improve some results on quasi-Frobenius rings.

Let $R$ be a ring and let $M$ be an $R$-module with $S={\mathrm{End}}_{\mathrm{R}}\left(\mathrm{M}\right)$. Consider the preradical $\overline{Z}$ for the category of right $R$-modules Mod-$R$ introduced by Y. Talebi and N. Vanaja in 2002 and defined by $\overline{Z}\left(M\right)=\bigcap \{U\le M:M/U$ is small in its injective hull$\}$. The module $M$ is called quasi-t-dual Baer if ${\sum }_{\varphi \in \Im }\varphi \left({\overline{Z}}^2\left(M\right)\right)$ is a direct summand of $M$ for every two-sided ideal $\Im$ of $S$, where ${\overline{Z}}^2\left(M\right)=\overline{Z}\left(\overline{Z}\left(M\right)\right)$. In this paper, we show that $M$ is quasi-t-dual Baer if and only if ${\overline{Z}}^2\left(M\right)$ is a direct summand of $M$ and ${\overline{Z}}^2\left(M\right)$ is a quasi-dual Baer module. It is also shown that any direct summand of a...

Let $R$ be a ring with identity and $M$ be a unitary left $R$-module. The co-intersection graph of proper submodules of $M$, denoted by $\Omega \left(M\right)$, is an undirected simple graph whose vertex set $V\left(\Omega \right)$ is a set of all nontrivial submodules of $M$ and two distinct vertices $N$ and $K$ are adjacent if and only if $N+K\ne M$. We study the connectivity, the core and the clique number of $\Omega \left(M\right)$. Also, we provide some conditions on the module $M$, under which the clique number of $\Omega \left(M\right)$ is infinite and $\Omega \left(M\right)$ is a planar graph. Moreover, we give several...

A ring is said to be strongly right bounded if every nonzero right ideal contains a nonzero ideal. In this paper strongly right bounded rings are characterized, conditions are determined which ensure that the split-null (or trivial) extension of a ring is strongly right bounded, and we characterize strongly right bounded right quasi-continuous split-null extensions of a left faithful ideal over a semiprime ring. This last result partially generalizes a result of C. Faith concerning split-null extensions...

In the first part, we study algebras A such that A = R ⨿ I, where R is a subalgebra and I a two-sided nilpotent ideal. Under certain conditions on I, we show that A is standardly stratified if and only if R is standardly stratified. Next, for $A=\left[\begin{array}{cc}U& 0\\ M& V\end{array}\right]$, we show that A is standardly stratified if and only if the algebra R = U × V is standardly stratified and ${}_{V}M$ is a good V-module.

In this paper, we prove that any pure submodule of a strict Mittag-Leffler module is a locally split submodule. As applications, we discuss some relations between locally split monomorphisms and locally split epimorphisms and give a partial answer to the open problem whether Gorenstein projective modules are Ding projective.

Let Λ be an artinian ring and let 𝔯 denote its Jacobson radical. We show that a simple module of finite projective dimension has no self-extensions when Λ is graded by its radical, with at most two simple modules and 𝔯⁴ = 0, in particular, when Λ is a finite-dimensional algebra over an algebraically closed field with at most two simple modules and 𝔯³ = 0.

An exchange ring $R$ is strongly separative provided that for all finitely generated projective right $R$-modules $A$ and $B$, $A\oplus A\cong A\oplus B\Rightarrow A\cong B$. We prove that an exchange ring $R$ is strongly separative if and only if for any corner $S$ of $R$, $aS+bS=S$ implies that there exist $u,v\in S$ such that $au=bv$ and $Su+Sv=S$ if and only if for any corner $S$ of $R$, $aS+bS=S$ implies that there exists a right invertible matrix $\left(\begin{array}{cc}a& b\\ & *\end{array}\right)\in M_2\left(S\right)$. The dual assertions are also proved.