**NAZIM AGAYEV**, **SAIT HALICIOĞLU ** and **ABDULLAH HARMANCI**

*On symmetric modules*

**Pages** 91-99

**Received:** 18 March 2009

**Accepted in revised form:** 19 May 2009

**Mathematics Subject Classification (2000):** 16U80

**Abstract**
Let *α* be an endomorphism of an arbitrary ring *R* with identity and let *M*
be a right *R*-module. We introduce the notion of *α*-symmetric modules as a
generalization of *α*-reduced modules. A module *M* is called *α-symmetric* if,
for any *m* ∈ *M* and any *a*, *b* ∈ *R*, *mab* = 0 implies *mba* = 0; *ma* = 0 if
and only if *mα*(*a*) = 0. We show that the class of *α*-symmetric modules lies
strictly between classes of *α*-reduced modules and α-semicommutative modules.
We study characterizations of *α*-symmetric modules and their related properties
including module extensions. For a rigid module *M*, *M* is *α*-reduced if and only
if *M* is *α*-symmetric. For a module *M*, it is proved that *M*[*x*]* _{R}*[

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