Analogues of a theorem of Cohen for overrings

Abstract

An integral domain R has the finite [finite presentation] overring property, if every overring of R in the quotient field K of R is a finitely generated [finitely presented] R-algebra. Papick in [24] (Proposition 23) has given a characterization of one-dimensional domains with the finite presentation overring property. In this paper, we remove the restriction on the dimension. Thri is achieved in Theorem 18 using a characterization of domains with the finite overring property (Theorem 14). An important contribution in obtaining these results comes from a consideration of an analogue for overrings of the following theorem of Cohen [17, Theorem 71. Let 1 be an ideal of a commutative unitary ring R. Assume that I is not finitely generated and is maximal with respect to this property. Then I is a prime ideal of R. Studies on overrings of integral domains [ 12,13,14,18,28] suggest that, in multiplicative domains, overrings of domains play a role analogous to ideals in commutative rings and that valuation overrings may be thought of as analogues of prime ideals. Looking for an analogue of the above theorem of Cohen, we are led to introduce the following definition: Let R be a G-domain, with quotient field K. We say that an overring S of R in K is a Cohen overring of R, if S is not a finitely generated R-algebra and S is maximal with respect to this property. We show that Cohen overrings are always local rings (Proposition 2). We exhibit examples showing that they may not be valuation rings. However, we point out that there are