$r$-Submodules and $uz$-modules

Document Type : Research Paper


Department of Mathematics, Faculty of Mathematical Sciences and Computer, Shahid Chamran University of Ahvaz, Ahvaz, Iran.



In this article we study and investigate the behavior of $r$-submodules (a proper submodule $N$ of an $R$-module $M$  in which $am\in N$ with ${\rm Ann}_M(a)=(0)$ implies that $m\in N$ for each $a\in R$ and $m\in M$). We show that every simple submodule, direct summand,  divisible submodule, torsion submodule and the socle of a module is an $r$-submodule and if $R$ is a domain, then the singular submodule is an $r$-submodule. We also introduce the concepts of $uz$-module (i.e., an $R$-module $M$ such that either ${\rm Ann}_M(a)\not=(0)$ or $aM=M$, for every $a\in R$) and strongly  $uz$-module (i.e., an $R$-module $M$ such that $aM\subseteq a^2M$, for every $a\in R$) in the category of modules over commutative rings. We show that every Von Neumann regular module is a strongly $uz$-module and every Artinian  $R$-module is a   $uz$-module. It is observed that if $M$ is a faithful cyclic $R$-module, then  $M$ is a   $uz$-module if and only if every its cyclic submodule is an $r$-submodule. In addition, in this case, $R$ is a domain if and only if the only $r$-submodule of $M$ is zero submodule. Finally, we prove that $R$  is a $uz$-ring if and only if every faithful cyclic $R$-module is a $uz$-module.


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