On Crossed Product Rings Over p.q.-Baer and Quasi-Baer Rings

Abstract

In this paper, we consider a ring R and a monoid M equipped with a twisting map f: M×M -> U(R) and an action map ω: M -> Aut(R). The main objective of our study is to investigate the conditions under which the crossed product structure R⋊M is p.q.-Baer and quasi-Baer rings, and how this property relates to the p.q.-Baer property of R and the existence of a generalized join in I(R) for M-indexed subsets, where I(R) denotes the set of ideals of R. Additionally, we prove a connection between R being a left p.q.-Baer ring and the CM-quasi-Armendariz property. Moreover, we prove that for any element φ2=φ, there exist an idempotent element e2=e such that φ = ce. We then prove that R is quasi-Baer if and only if the crossed product structure R⋊M is quasi-Baer. Finally, we present novel results regarding various constructions for crossed products.

Some Results of Malcev-Neumann Rings

“…This construction has appeared in many papers, mainly in the study various properties of division rings and related topic. For a more comprehensive understanding of this construction and the results associated with it, it is recommended to refer to several scholarly papers on the topic [3], [6], [9], [10], [15] and [16]. The subring of R * ((G)) consisting of all finite sums f = x∈G r x x (i.e., sums of finite support) is just the twisted group ring R * (G).…”