Zero Insertive Nil Clean Rings

TL;DR

This paper explores the properties of zero insertive nil clean (ZINC) rings, characterizing their relationships with semicommutative rings, their behavior under direct products, and conditions for matrix and extension rings to be ZINC.

Contribution

It introduces the concept of zero insertive nil clean rings, establishes their key properties, and analyzes their behavior under various algebraic constructions.

Findings

ZINC rings relate to semicommutative and weakly semicommutative rings.

Finite products of ZINC rings are ZINC if each component is ZINC.

Matrix rings over ZINC rings are ZINC only under specific conditions.

Abstract

This paper investigates key properties of ZINC rings and their relationships with semicommutative and weakly semicommutative rings. We call an element $x$ of a ring $R$ zero insertive if $x=arb$ for some $a,b,r\in R$ such that $ab=0$ and $ZI(R)$ denotes the set of all zero insertive elements of $R$. We establish that a ring $R$ is semicommutative if and only if $ZI(R) \subseteq E(R)$, and weakly semicommutative if and only if $ZI(R) \subseteq N(R)$, where $E(R)$ and $N(R)$ denote respectively the sets of idempotent elements and nilpotent elements. For ZINC rings with no nontrivial idempotents, $ZI(R) \subseteq N(R)$. We prove that a finite direct product of ZINC rings is ZINC if and only if each component ring is ZINC, while an infinite direct product may fail to be ZINC. For $n \geq 2$, if $M_n(R)$ is ZINC, then $R$ is weakly clean, however, the converse is not true (e.g., $\mathbb{Z}$). Additionally, $M_n(K)$ is ZINC for a division ring $K$ if and only if $K \cong \mathbb{F}_2$. We, also, present a ZINC ring whose polynomial and power series extensions are not ZINC.