Commutative rings whose proper homomorphic images are nil clean

Mahdi Samiei

As defined by Diesl a (noncommutative) ring $R$ is called nil clean if every element of $R$ is a sum of a nilpotent and an idempotent. The purpose of this paper is to study and investigate a new class of rings called nil neat rings, which is presented in \cite[Problem $4$]{comm.weaklynilclean}. Actually, these rings are a natural generalization of the notion of neat rings, as rings for which any proper homomorphic images are nil clean. It is well-known that any homomorphic image of a nil clean ring is again nil clean. In this paper, it is proved that a nil neat ring which is not nil clean is either a field that is not isomorphic to $\mathbb{Z}_2$ or a one-dimensional domain. We also show that a ring $R$ is nil neat if and only if every nonzero prime ideal of $R$ is maximal, and that for all nonzero maximal ideals $M$ of $R$, $R/M\cong \mathbb{Z}_2$.