Integral domains which are almost Dedekind

Abstract

An integral domain J (with unit) will be said to be almost Dedekind if, given any maximal ideal P of J, Jp is a Dedekind domain. It follows, in particular, that Jp is a discrete valuation ring. I. S. Cohen, in [1, p. 33] gives six necessary and sufficient conditions that a Noetherian domain J be Dedekind. The first of these is that Jp be a discrete valuation ring for every maximal ideal P of J. This paper gives necessary and sufficient conditions that an integral domain with unit be almost Dedekind and it is shown that these conditions imply each of the other five conditions of Cohen. Finally, we investigate relations between the ideal structure of D' and D, where D is an almost Dedekind domain with quotient field K and DCD'CK. In particular, we show D' is almost Dedekind and that D' is the intersection of all quotient rings Dp of D where P is a prime ideal of D such that PD' CD'. The results obtained also yield another proof to the theorem of MacLane and Schilling [2, p. 781] which asserts that if D is Dedekind, so is D'.