Some Results of Malcev-Neumann Rings

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DOI: https://doi.org/10.28924/2291-8639-22-2024-39
Published: Feb 27, 2024
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Kholood Alnefaie, Eltiyeb Ali, Some Results of Malcev-Neumann Rings, Int. J. Anal. Appl., 22 (2024), 39.

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Kholood Alnefaie, Eltiyeb Ali

Abstract

Let us consider the function σ, which maps elements from the group G to the group of automorphisms of the ring R. In this paper, we are studying new conditions under which the Malcev-Neumann ring R∗((G)) is a PS, APP, PF, PP, and a Zip rings, respectively. It has been demonstrated that if R is a reduced ring and σ is weakly rigid, then the Malcev-Neumann ring R∗((G)) over a PS-ring is a PS. Furthermore, if σ is weakly rigid and the ring R satisfies the descending chain condition on left annihilators, then the Malcev-Neumann ring R∗((G)) is a right APP-ring if and only if, for any G-indexed generated right ideal A of R, rR(A) is left s-unital. Additionally, we have proven that if R is a commutative ring and σ is weakly rigid, then the Malcev-Neumann ring R∗((G)) is a PF ring if and only if, for any two G-indexed subsets A and B of R such that B⊆annR(A), there exists c∈annR(A) such that bc = b for all b ∈ B. These results extend the corresponding findings for polynomial rings and Laurent power series rings.

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