  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.

Rings and Algebras