  In this paper, we introduce and study a strict generalization of symmetric
rings. We call a ring $R \,\,\, 'P-symmetric'$ 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.
