Commutative Weakly Invo–Clean Group Rings

Peter V Danchev

doi:10.15826/umj.2019.1.005

Abstract

A ring \(R\) is called weakly invo-clean if any its element is the sum or the difference of an involution and an idempotent. For each commutative unital ring \(R\) and each abelian group \(G\), we find only in terms of \(R\), \(G\) and their sections a necessary and sufficient condition when the group ring \(R[G]\) is weakly invo-clean. Our established result parallels to that due to Danchev-McGovern published in J. Algebra (2015) and proved for weakly nil-clean rings.

Highlights

Introduction and conventionsThroughout the current paper, we shall assume that all rings R are associative, containing the identity element 1 which differs from the zero element 0
By the usage of Lemma 1 (ii) above, we derive that R2[G] is an invo-clean ring of characteristic 3
It is worthwhile noticing that concrete examples of an invo-clean ring of characteristic 4, such that its elements are solutions of the equation 2r2 = 2r, are the rings Z4 and Z4 × Z4

Summary

Introduction

Introduction and conventionsThroughout the current paper, we shall assume that all rings R are associative, containing the identity element 1 which differs from the zero element 0. If r = v + e or r = v − e, the ring is called weakly invo-clean. A criterion for an arbitrary commutative group ring to be nil-clean was recently obtained in [8].

Results

Conclusion