Structure of a factor ring $R/P$ in terms of differential identities involving a new kind of involution

TL;DR

This paper investigates the structure of factor rings formed by prime ideals in rings, using derivations and a novel involution to establish criteria for their commutativity.

Contribution

It introduces a new type of involution and links derivations satisfying algebraic identities to the commutativity of factor rings.

Findings

Criteria for commutativity of $R/P$ based on derivations

Introduction of a new involution related to prime ideals

Conditions under which $R/P$ becomes commutative

Abstract

Let $R$ be a ring and $P$ a prime ideal of $R.$ In this paper, we establish some commutativity criteria for the factor ring $R/P$ in terms of derivations of $R$ satisfying some algebraic identities involving a new kind of involution in relation with $P.$