Abstract:  
The behavior of a ring is governed, to some degree, by its abundance (or lack) or certain special kinds of elements (e.g. units, idempotents, nilpotents).  More broadly, one can ask about the degree to which an arbitrary element of a ring can be expressed as a sum of two (or more) of these special elements.  This philosophy has motivated the development and study of a variety of classes of rings, defined by conditions of this type.  For example, a ring is called clean if every element can be expressed as a sum of an idempotent and a unit.  In this talk, I will begin with some background and history about conditions of this type before moving into a discussion of some recent and ongoing work into ideal-theoretic versions of these conditions.  Specifically, I will talk about my work (joint with Alexi Block Gorman and Tom Dorsey) on a new class of rings called "Ideally Nil Clean" rings, defined by the property that every ideal is a sum of an idempotent ideal and a nil ideal.  No background beyond some basics of ring theory will be assumed.