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

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 $$\mathrm{char}(R_{2})=3$$.

### MSC:

 16S34 Group rings 16U99 Conditions on elements

### Keywords:

invo-clean rings; weakly invo-clean rings; group rings
Full Text:

### References:

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