LNZS Rings

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.