Some Results of Malcev-Neumann Rings

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.