CHARACTERIZATIONS OF $t^2$-REVERSIBLE RINGS

This article aims to investigate the ring theoretic structures of (strongly) $t^2$-reversible ring using the concept of non-zero tripotent elements. A ring $R$ is said to be $t^2$-reversible if $ab=0$ implies $bat^2=0$ for all $a,b\in R$ and $t$ is a non-zero tripotent element of $R$. It is proved that $R$ is a $t^2$-reversible ring if and only if $t^2$ is left semicentral and $t^2Rt^2$ is a reversible ring. We also introduce and establish several characteristics of strongly $t^2$-reversible rings. It is proved that every strongly $t^2$-reversible ring is also a $t^2$-reversible ring but the converse need not be true. Moreover we call, $R$ is a right (left) $t^2$-reduced ring if $N(R)t^2=0$ $(t^2N(R)=0)$, where $N(R)$ stands for the set of all nilpotent elements of $R$ and we have established some of its properties.