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...

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.