Ideally nil clean rings

Abstract

A ring is called nil clean if every element can be written as the sum of an idempotent and a nilpotent. The class of nil clean rings has emerged as an important variant of the much-studied class of clean rings. However, the nil clean definition is fairly restrictive, and the collection of examples is rather constrained. In this paper, we propose an ideal-theoretic version of the nil clean concept, with the goal of creating a more expansive class of rings. Specifically, we define a ring to be ideally nil clean (INC) if every (two-sided) ideal can be written as the sum of an idempotent ideal and a nil ideal. By passing from elements to ideals, we vastly expand the collection of rings under consideration. For example, all artinian rings and all von Neumann regular rings are INC. We show that, in the commutative case, the INC rings are exactly the strongly π-regular rings. We also explore the relationship between the nil clean and INC properties. After establishing a robust collection of examples of INC rings, we explore the behavior of the INC condition under common ring extensions. In addition, we state several open questions and suggest areas for further investigation.