Nil-reversible rings

TL;DR

This paper introduces nil-reversible rings, a new class where the annihilators of nilpotent elements are equal, explores their properties, and examines conditions under which polynomial rings over them are also nil-reversible.

Contribution

The paper defines nil-reversible rings, studies their properties, and establishes conditions for polynomial rings over them to retain nil-reversibility, expanding the understanding of ring structures.

Findings

Nil-reversible rings are abelian, 2-primal, weakly semicommutative, and nil-Armendariz.

Reversible rings are a proper subclass of nil-reversible rings.

Polynomial rings over nil-reversible rings are not necessarily nil-reversible unless the ring is Armendariz.

Abstract

This paper introduces and studies nil-reversible rings wherein we call a ring R nil-reversible if the left and right annihilators of every nilpotent element of R are equal. Reversible rings (and hence reduced rings) form a proper subclass of nil-reversible rings and hence we provide some conditions for a nil-reversible ring to be reduced. It turns out that nil-reversible rings are abelian, 2-primal, weakly semicommutative and nil-Armendariz. Further, we observe that the polynomial ring over a nil-reversible ring R is not necessarily nil-reversible in general, but it is nil-reversible if R is Armendariz additionally.