LNZS Rings

TL;DR

This paper studies a new class of rings called LNZS rings, exploring their properties, relationships with reduced rings, and how polynomial and skew polynomial extensions affect the LNZS property.

Contribution

It introduces LNZS rings, characterizes when they are reduced, and examines the LNZS property in polynomial and skew polynomial extensions.

Findings

Reduced rings are LNZS but not vice versa.

R is reduced iff the 2x2 upper triangular matrix ring over R is LNZS.

Polynomial and skew polynomial rings over LNZS rings are generally not LNZS.

Abstract

This paper introduces a class of rings called left nil zero semicommutative rings ( LNZS rings ), wherein a ring R is said to be LNZS if the left annihilator of every nilpotent element of R is an ideal of R. It is observed that reduced rings are LNZS but not the other way around. So, this paper provides some conditions for an LNZS ring to be reduced, and among other results, it is proved that R is reduced if and only if the ring of upper triangular matrices of order 2x2 over R is LNZS. Furthermore, it is shown through examples that neither the polynomial ring nor the skew polynomial ring over an LNZS is LNZS. Therefore, this paper investigates the LNZS property of the polynomial extension and skew polynomial extension of an LNZS ring with some additional conditions.