A STUDY ON $t^2$-SYMMETRIC RINGS

In this article our attempt is to study the ring theoretic properties of $t^2$-symmetric and strongly $t^2$-symmetric rings of tripotent elements of a ring. Let $R$ be a ring and $t$ be a tripotent element of $R$, then $R$ is said to be $t^2$-symmetric if $abc=0$ implies $acbt^2=0$ for all $a,b,c\in R$. It has been proved that $R$ is a $t^2$-symmetric ring if and only if $t^2$ is left semicentral and $t^2Rt^2$ is a symmetric ring. We also introduce the strongly $t^2$-symmetric ring and establish various properties of it.