ABSTRACT
In this paper we will briefly survey results concerning "pairs" of rings possessing certain algebraic prpperties, especially finiteness conditions. Our focus sets the stage for a concise discussion of some seminal work of Gilmer and Heinzer concerning special affine extensions of commutative rings. In particular, we will highlight a variety of their significant results pertaining to such extensions and indicate some important connections to the work of others. The treatment will be quite leisurely and certainly not totally exhaustive. A convention throughout this manuscript is that, unless otherwise stated, all rings are assumed to be integral domains that are not fields.
