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.

摘要

如果每个元素都可以写为幂等和幂等的和，则环称为零净。零净环类已成为广泛研究的净环类的一个重要变体。然而，nil clean 的定义相当严格，示例的集合也相当有限。在本文中，我们提出了一个理想理论版本的 nil clean 概念，目的是创建一个更广泛的环类。具体来说，如果每个（两侧）理想都可以写为幂等理想和零理想之和，我们将环定义为理想零净（INC）。通过从元素到理想的传递，我们极大地扩展了正在考虑的戒指系列。例如，所有亚丁环和所有冯诺依曼正则环都是 INC。我们证明，在交换情况下，π - 规则环。我们还探讨了 nil clean 和 INC 属性之间的关系。在建立了一个强大的 INC 环示例集合之后，我们探索了 INC 条件在常见环扩展下的行为。此外，我们陈述了几个悬而未决的问题并建议了进一步调查的领域。