GENERALIZED SEMI COMMUTATIVE RINGS AND THEIR EXTENSIONS

Abstract

For an endomorphism <TEX>${\alpha}$</TEX> of a ring R, the endomorphism <TEX>${\alpha}$</TEX> is called semicommutative if ab=0 implies <TEX>$aR{\alpha}(b)$</TEX>=0 for a <TEX>${\in}$</TEX> R. A ring R is called <TEX>${\alpha}$</TEX>-semicommutative if there exists a semicommutative endomorphism <TEX>${\alpha}$</TEX> of R. In this paper, various results of semicommutative rings are extended to <TEX>${\alpha}$</TEX>-semicommutative rings. In addition, we introduce the notion of an <TEX>${\alpha}$</TEX>-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 <TEX>$R[[x;\;{\alpha}]]$</TEX>. We show that a number of interesting properties of a ring R transfer to its the skew power series ring <TEX>$R[[x;\;{\alpha}]]$</TEX> and vice-versa such as the Baer property and the p.p.-property, when R is <TEX>${\alpha}$</TEX>-skew power series Armendariz. Several known results relating to <TEX>${\alpha}$</TEX>-rigid rings can be obtained as corollaries of our results.