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Let $R$ be a commutative semiring with non-zero identity. In this paper, we introduce and study the graph $\Omega \left(R\right)$ whose vertices are all elements of $R$ and two distinct vertices $x$ and $y$ are adjacent if and only if the product of the co-ideals generated by $x$ and $y$ is $R$. Also, we study the interplay between the graph-theoretic properties of this graph and some algebraic properties of semirings. Finally, we present some relationships between the zero-divisor graph $\Gamma \left(R\right)$ and $\Omega \left(R\right)$.

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.

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...

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 $M$ be a module and $\mu$ be a class of modules in $Mod-R$ which is closed under isomorphisms and submodules. As a generalization of essential submodules Özcan in [8] defines a $\mu$-essential submodule provided it has a non-zero intersection with any non-zero submodule in $\mu$. We define and investigate $\mu$-singular modules. We also introduce $\mu$-extending and weakly $\mu$-extending modules and mainly study weakly $\mu$-extending modules. We give some characterizations of $\mu$-co-H-rings by weakly $\mu$-extending modules. Let $R$...