On Weakly Présimplifiable Group Rings

Abstract

A commutative ring R with unity is called weakly-pr´esimplifiable (resp.,pr´esimplifiable) if for a, b ∈ R with a = ba, then either a = 0 or b is a regular element (that is, b is not a zero-divisor) in R (resp., a = 0 or b is a unit in R). Let R be a commutative ring with unity and G be a nontrivial abelian group. In this paper, we give some characterizations for the group ring R[G] to be weakly pr´esimplifiable.
Furthermore, we give a complete description of (weakly) pr´esimplifiable circulant matrix ring.

Key words and phrases. Pr´esimplifiable ring, indecomposable ring, circulant matrix ring.

2020 Mathematics Subject Classification. 13A05, 16S34