Bull. Korean Math. Soc. 2018 Vol. 55, No. 4, 1179-1187 https://doi.org/10.4134/BKMS.b170656 Published online March 14, 2018 Printed July 1, 2018

Simion Breaz, Andrada Cimpean Babes-Bolyai University, Babes-Bolyai University

Abstract : We study the class of rings $R$ with the property that for $x\in R$ at least one of the elements $x$ and $1+x$ are tripotent. We prove that a commutative ring has this property if and only if it is a subring of a direct product $R_0\times R_1\times R_2$ such that $R_0/J(R_0)\cong \mathbb Z_2$, for every $x\in J(R_0)$ we have $x^2=2x$, $R_1$ is a Boolean ring, and $R_3$ is a subring of a direct product of copies of $\mathbb Z_3$.