**Authors:** Peter V. Danchev

### Abstract

Very recently, Breaz and C\^{\i}mpean introduced and examined in Bull. Korean Math. Soc. (2018) the
class of so-called {\it weakly tripotent rings} as those rings $R$ whose elements satisfy at leat one of the
equations $x^3=x$ or $(1-x)^3=1-x$. These rings are generally non-commutative. We here obtain a criterion when
the commutative group ring $RG$ is weakly tripotent in terms only of a ring $R$ and of a group $G$ plus their
sections. \newline
Actually, we also show that these weakly tripotent rings are {\it strongly invo-clean rings} in the sense of
Danchev in Commun. Korean Math. Soc. (2017). Thereby, our established criterion somewhat strengthens previous
results on commutative strongly invo-clean group rings, proved by the present author in Univ. J. Math.
\& Math. Sci. (2018). Moreover, this criterion helps us to construct a commutative strongly invo-clean ring
of characteristic $2$ which is {\it not} weakly tripotent, thus showing that these two ring classes are
different.

Institute of Mathematics and Informatics

Bulgarian Academy of Sciences

”Acad. G. Bonchev”, str., bl. 8, 1113 Sofia

Bulgaria

E-mail: ,

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