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Primary Wellesley Thesis Advisor
The notion of a clean ring has many variations that have been widely studied, including the sub-class of nil clean rings. We develop new variations of this concept and discuss the interactions between these new properties and those in the established canon. The first property we define is an ideal-theoretic generalization of the element-wise defined property "nil clean," the condition that an element of a ring is the sum of a nilpotent and an idempotent. We establish a few characterizations for certain families of rings with this property, called ideally nil clean. In particular, a commutative ring is ideally nil clean if and only if it is strongly pi-regular. We show that the class of ideally nil clean rings also includes artinian rings, and von Neumann regular rings. Among other results, we demonstrate that ideally nil clean rings behave well under some ring extensions such as direct products and matrix rings. We also expand this generalization to the ideally nil clean property for one-sided ideals, and discuss the interaction between these different generalizations. We explore the interplay between nil clean rings and ideally nil clean rings.