Residual finiteness in commutative rings: structure, stability, and local variants

This paper studies residually finite rings, a class of commutative rings in which every nonzero ideal has a finite quotient. We begin by examining several properties of residually finite rings, with particular emphasis on how residual finiteness is preserved under distinguished ring extensions, including direct products, flat overrings, finite extensions, polynomial and power series rings, as well as Nagata and Anderson rings. We also provide a negative answer to an open question posed by Ion and Militaru. Finally, we introduce and investigate a proper generalization of this concept, which we call locally residually finite rings.