Symmetricity of rings relative to the prime radical

In this paper, we introduce and study a strict generalization of symmetric rings. We call a ring $R{}^{\mathrm{\prime }}P-symmetri{c}^{\mathrm{\prime }}$ if for any $a,, b,, c\in R,, abc=0$implies$bac\in P®$,where$P®$istheprimeradicalof$R$. 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.