Maximal Non-Noetherian Subrings of a Domain

The main purpose of this paper is to study maximal non-Noetherian subrings R of a domain S. We give characterizations of such domains in several cases. If the ring R is semi-local, (R, S) is a residually algebraic pair and R is a maximal non-Noetherian subring of S, we give sharp upper bounds for the number of rings and the length of chains of rings in [R, S], the set of intermediary rings.