Zero Insertive Nil Clean Rings

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.