On $p-$Ring

Abstract

In this paper, we introduced the concept of a $p$-ideal for a given ring. We provide necessary and sufficient condition for $\dfrac{R[x]}{(f(x))}$ to be a $p$-ring, where $R$ is a finite $p$-ring. It is also shown that the amalgamation of rings, $A\bowtie^fJ$ is a $p$-ring if and only if so is $A$ and $J$ is a $p$-ideal. Finally, we establish the transfer of this notion to trivial ring extensions.