Generalized semicommutative rings and their extensions

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