Generalized semicommutative rings and their extensions

Abstract

For an endomorphism a of α ring R, the endomorphism α is called semicommutative if ab - 0 implies a Rα(b) = 0 for a ∈ R. A ring R is called α-semicommutative if there exists a semicommutative endomorphism α of R. In this paper, various results of semicommutative rings are extended to α-semicommutative rings. In addition, we introduce the notion of an α-skew power series Armendariz ring which is an extension of Armendariz property in a ring R by considering the polynomials in the skew power series ring R x; α]]. We show that a number of interesting properties of a ring R transfer to its the skew power series ring R x; α]] and vice-versa such as the Baer property and the p.p.-property, when R is α-skew power series Armendariz. Several known results relating to α-rigid rings can be obtained as corollaries of our results. © 2008 The Korean Mathematical Society.