Bulletin of the
Korean Mathematical Society BKMS

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

MSC numbers: 16U80, 16W20, 16W60