Generalized Baer and Generalized Quasi-Baer Rings of Skew Generalized Power Series

TL;DR

This paper investigates the properties of skew generalized power series rings over generalized Baer and quasi-Baer rings, establishing conditions under which these properties are preserved in the extension.

Contribution

It extends the theory of generalized Baer and quasi-Baer rings to skew generalized power series rings, providing necessary and sufficient conditions for property preservation.

Findings

The ring $A$ is generalized right Baer if and only if $R$ is generalized right Baer under certain conditions.

The ring $A$ is generalized right quasi-Baer if and only if $R$ is generalized right quasi-Baer under certain conditions.

The paper characterizes how these properties transfer between $R$ and $A$ in the context of skew generalized power series.

Abstract

Let $R$ be a ring with identity, $(S,\leq)$ an ordered monoid, $\omega:S \to End(R)$ a monoid homomorphism, and $A= R\left[\left[S,\omega \right]\right]$ the ring of skew generalized power series. The concepts of generalized Baer and generalized quasi-Baer rings are generalization of Baer and quasi-Baer rings, respectively. A ring $R$ is called generalized right Baer (generalized right quasi-Baer) if for any non-empty subset $S$ (right ideal $I$) of $R$, the right annihilator of $S^n \space{0.1cm}(I^n)$ is generated by an idempotent for some positive integer $n$. Left cases may be defined analogously. A ring $R$ is called generalized Baer (generalized quasi-Baer) if it is both generalized right and left Baer (generalized right and left quasi-Baer) ring. In this paper, we examine the behavior of a skew generalized power series ring over a generalized right Baer (generalized right quasi-Baer) ring and prove that, under specific conditions, the ring $A$ is generalized right Baer (generalized right quasi-Baer) if and only if $R$ is a generalized right Baer (generalized right quasi-Baer) ring.