Special Properties of Rings of Crossed Products Type
DOI:
https://doi.org/10.29020/nybg.ejpam.v19i2.7713Keywords:
Clean ring; Dedekind finite ring; Exchange ring; CM-Armendariz ring; Crossed products of M over a ring RAbstract
Let R be a ring, M a strictly ordered monoid with a twisting map f : M×M →U(R) and an action map ω : M →Aut(R). In this paper, we introduce the concept of strongly CM-reversible rings and investigates the properties of crossed products of R with M, focusing on the CM-Armendariz condition and its extension from polynomial rings to these crossed products. It is shown that R∗M is semicommutative (symmetric, Dedekind finite, clean, uniquely clean, or exchange) if and only if Ris semicommutative (symmetric, Dedekind finite, clean, uniquely clean, or exchange ring, respectively) under some additional conditions. Additionally, we prove that a ring R is strongly CM-reversible if and only if it is a right Ore ring with a classical right quotient ring Q. Finally, we discuss various properties related to strongly CM-reversibility.
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Copyright (c) 2026 Awn Alqahtani
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