On $p-$Ring

TL;DR

This paper introduces the concept of p-ideals in rings, characterizes when polynomial quotient rings are p-rings, and explores how this property transfers through ring constructions like amalgamations and trivial extensions.

Contribution

It defines p-ideals, provides necessary and sufficient conditions for polynomial quotient rings to be p-rings, and studies the transfer of p-ring properties in ring constructions.

Findings

Characterization of when R[x]/(f(x)) is a p-ring for finite p-rings R.

p-ring property transfer in amalgamation of rings.

Transfer of p-ring property to trivial ring extensions.

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.