Symmetricity of rings relative to the prime radical

Abstract

In this paper, we introduce and study a strict generalization of symmetric rings. We call a ring $R{}^{\prime }P-symmetri{c}^{\prime }$ if for any $a,b,c\in R,abc=0$ implies $bac\in P(R)$, where $P(R)$ is the prime radical of $R$. It is shown that the class of $P$-symmetric rings lies between the class of central symmetric rings and generalized weakly symmetric rings. Relations are provided between $P$-symmetric rings and some other known classes of rings. From an arbitrary $P$-symmetric ring, we produce many families of $P$-symmetric rings.