Analogues of a theorem of Cohen for overrings

The Space of Maximal Subrings of a Commutative Ring

“…In what follows, we present a sequence of results which supports the latter comment in the introduction of [16]. First we need the following interesting fact which is in [23, Theorem 107].…”

“…Before going any further and for an example and an application of the previous facts, let us review an amazing and interesting result due to I. J. Papick, which reminds one of the well-known result of Anderson on finiteness of minimal prime ideals of a commutative rings, see [1]. A special case of the following fact is proved in [16,Theorem 1], for a G-domain R with quotient field K. Then one can conclude that if all the valuation overrings of R are finitely generated as rings over R, then the integral closure of R in K is a strong G-domain. Fortunately its proof, which is by invoking of Zariski-Riemann surface, is valid for our claim for any integral domain.…”

“…As mentioned in the introduction of [16], the situation of overrings of an integral domain and the valuation overrings of its quotient fields are very similar to ideals and prime ideals of a ring. In what follows, we present a sequence of results which supports the latter comment in the introduction of [16].…”