Commutative weakly invo-clean group rings. (English) Zbl 1465.16020

Danchev, Peter V.

Ural Math. J. 5, No. 1, 48-52 (2019).

A ring \(R\) is said to be invo-clean if, for every \(r\in R\), there exist an involution \(v\) and an idempotent \(e\) such that \(r=v+e\). If \(r=v+e\) or \(r=v-e\), the ring is called weakly invo-clean.
The following result is proved as
Theorem 1. Let \(G\) be an abelian group and let \(R\) be a commutative non-zero ring. Then the group ring \(R[G]\) is weakly invo-clean if and only if at most one of the following is true:
(1) \(G=\{1\}\) and \(R\) is weakly invo-clean.
(2) \(G\neq \{1\}\) and \(R\cong R_{1}\times R_{2}\) is invo-clean such that either
(2.1) \(\left\vert G\right\vert >2\), \(G^{2}=\{1\}\), \(R_{1}=\{0\}\) or \(R_{1}\) is a ring of \(\mathrm{char}(R_{1})=2\), and \(R_{2}=\{0\}\) or \(R_{2}\) is a ring of \(\mathrm{char}(R_{2})=3\)
or
(2.2) \(\left\vert G\right\vert =2\), \(2r_{1}^{2}=2r_{1}\) for all \(r_{1}\in R_{1}\) (in addition \(4=0\) in \(R_{1}\)), and \(R_{2}=\{0\}\) or \(R_{2}\) is a ring of \(\mathrm{char}(R_{2})=3\).

Reviewer: Grigore Călugăreanu (Cluj-Napoca)

MSC:

16S34	Group rings
16U99	Conditions on elements

Keywords:

invo-clean rings; weakly invo-clean rings; group rings

References:

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