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arXiv · 2405.03423

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

M. M. Hamam·R. E. Abdel-Khalek·R. M. Salem
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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.

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BibTeXRIS

M. M. Hamam, R. E. Abdel-Khalek, R. M. Salem. 2024-05-06. Generalized Baer and Generalized Quasi-Baer Rings of Skew Generalized Power Series. https://arxiv.org/abs/2405.03423

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