# On $\mu $-singular and $\mu $-extending modules

## Yahya Talebi and Ali Reza Moniri Hamzekolaee

Address: Department of Mathematics, Faculty of Mathematical Science, University of Mazandaran, Babolsar, Iran

E-mail:

talebi@umz.ac.ir

a.monirih@umz.ac.ir

Abstract: Let $M$ be a module and $\mu $ be a class of modules in $\operatorname{Mod}-R$ 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 $R$ be a right non-$\mu $-singular ring such that all injective modules are non-$\mu $-singular, then $R$ is right $\mu $-co-H-ring if and only if $R$ is a QF-ring.

AMSclassification: primary 16S90; secondary 16D10, 16D70, 16D99.

Keywords: $\mu$-essential submodule, $\mu$-singular module, $\mu$-extending module, weakly $\mu$-extending module.

DOI: 10.5817/AM2012-3-183