PS-Modules over Ore Extensions and Skew Generalized Power Series Rings

Refaat M Salem,Mohamed A Farahat,Hanan Abd-Elmalk

doi:10.1155/2015/879129

Abstract

A rightR-moduleMRis called a PS-module if its socle,SocMR, is projective. We investigate PS-modules over Ore extension and skew generalized power series extension. LetRbe an associative ring with identity,MRa unitary rightR-module,O=Rx;α,δOre extension,MxOa rightO-module,S,≤a strictly ordered additive monoid,ω:S→EndRa monoid homomorphism,A=RS,≤,ωthe skew generalized power series ring, andBA=MS,≤RS,≤, ωthe skew generalized power series module. Then, under some certain conditions, we prove the following: (1) IfMRis a right PS-module, thenMxOis a right PS-module. (2) IfMRis a right PS-module, thenBAis a right PS-module.

Highlights

We investigate PS-modules over Ore extension and skew generalized power series extension
Throughout this paper R denotes an associative ring with identity and MR a unitary right R-module
The motivation of this paper is to investigate the PS property of Ore extension modules and the skew generalized power series extension modules

Summary

Introduction

Throughout this paper R denotes an associative ring with identity and MR a unitary right R-module. A ring R is said to be a left PS-ring if RR is a PS-module. In particular every Baer ring is a PS-ring (where a ring R is called Baer if every left (or right) annihilator is generated by an idempotent). As a generalization of left duo rings, a ring R is called weakly left duo if for every r ∈ R there is a natural number n(r) such that Rrn(r) is a two-sided ideal of R. A ring R is weakly duo if it is weakly right and left duo. In [4], Dingguo generalized this result to weakly duo rings as follows: a weakly duo reduced ring R is a PS-ring if and only if R is a right PS-ring. These results generalize the corresponding results for polynomial rings, generalized power series rings, and modules [5, 6]

PS-Modules over Ore Extension Rings

PS-Modules over Skew Generalized Power Series Rings