Semicentral Idempotents Research Articles

Skew power series extensions of principally quasi-Baer rings

A ring R is called right principally quasi-Baer (or simply right p.q.-Baer ) if the right annihilator of a principal right ideal of R is generated by an idempotent. Let R be a ring such that all left semicentral idempotents are central. Let α be an endomorphism of R which is not assumed to be surjective and R be α -compatible. It is shown that the skew power series ring R [[ x; α ]] is right p.q.-Baer if and only if the skew Laurent power series ring R [[ x, x−1 ; α ]] is right p.q.-Baer if and only if R is right p.q.-Baer and any countable family of idempotents in R has a generalized join in I ( R ). An example showing that the α -compatible condition on R is not superfluous, is provided.

A NOTE ON EXTENSIONS OF PRINCIPALLY QUASI-BAER RINGS

Let $R$ be a ring with unity. It is shown that the formal power series ring $R[[x]]$ is right p.q.-Baer if and only if $R$ is right p.q.-Baer and every countable subset of right semicentral idempotents has a generalized countable join.

A NOTE ON PRINCIPALLY QUASI-BAER RINGS

ABSTRACT Let be a ring such that all left semicentral idempotents are central. It is shown that is right p.q.Baer if and only if is right p.q.Baer and any countable family of idempotents in has a generalized join in .

Algebras Generated by Semicentral Idempotents

Let \(R\) be a unital K-algebra, where K is a commutative ring with unity. An idempotent \(e \in R\) is {\it left semicentral\/} if \(Re = eRe\), and \(R\) is {\it SCI-generated\/} if it is generated as a K-module by left semicentral idempotents. This paper develops the basic properties of SCI-generated algebras and characterizes those that are also prime, semiprime, primitive, or subdirectly irreducible. Minimal ideals and the socle of SCI-generated algebras are investigated. Conditions are found to describe a large class of SCI-generated algebras via generalized triangular matrix representations. SCI-generated piecewise domains are characterized. Examples are given that illustrate the breadth and diversity of the class of SCI-generated algebras.