Nil-good and nil-good clean matrix rings

TL;DR

This paper introduces and explores new variations of clean and 2-good rings, specifically nil-good and nil-good clean rings, analyzing their properties and relationships within noncommutative algebra.

Contribution

It defines nil-good and nil-good clean rings, establishing their properties and connections to existing concepts in noncommutative algebra.

Findings

The endomorphism ring of a module over a division ring is nil-good.

Nil-good rings include certain classes of rings with specific element decompositions.

Nil-good clean rings generalize clean rings with additional nilpotent and idempotent conditions.

Abstract

The notion of clean rings and 2-good rings have many variations, and have been widely studied. We provide a few results about two new variations of these concepts and discuss the theory that ties these variations to objects and properties of interest to noncommutative algebraists. A ring is called nil-good if each element in the ring is the sum of a nilpotent element and either a unit or zero. We establish that the ring of endomorphisms of a module over a division is nil-good, as well as some basic consequences. We then define a new property we call nil-good clean, the condition that an element of a ring is the sum of a nilpotent, an idempotent, and a unit. We explore the interplay between these properties and the notion of clean rings.