Generalized semicommutative rings and their extensions

Bull. Korean Math. Soc. 2008 Vol. 45, No. 2, 285-297 Printed June 1, 2008

Muhittin Ba\c{s}er, Abdullah Harmanci, and Tai Keun Kwak Kocatepe University, Hacettepe University, and Daejin University

Abstract : For an endomorphism $\alpha$ of a ring $R$, the endomorphism $\alpha$ is called semicommutative if $ab=0$ implies $aR\alpha(b)=0$ for $a \in R$. A ring $R$ is called $\alpha$-semicommutative if there exists a semicommutative endomorphism $\alpha$ of $R$. In this paper, various results of semicommutative rings are extended to $\alpha$-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 $\alpha$-skew power series Armendariz. Several known results relating to $\alpha$-rigid rings can be obtained as corollaries of our results.

Keywords : semicommutative rings, rigid rings, skew power series rings, extended Armendariz rings, Baer rings, p.p.-rings