Generalized semicommutative rings and their extensions. (English) Zbl 1144.16025

Başer, Muhittin; Harmanci, Abdullah; Kwak, Tai Keun

Bull. Korean Math. Soc. 45, No. 2, 285-297 (2008).

Summary: 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.

MSC:

16U80	Generalizations of commutativity (associative rings and algebras)
16W20	Automorphisms and endomorphisms
16W60	Valuations, completions, formal power series and related constructions (associative rings and algebras)

Keywords:

semicommutative rings; semicommutative endomorphisms; rigid rings; skew power series rings; extended Armendariz rings; Baer rings; p.p.-rings