A valuation domain is an integral domain R such that for all a,b R either a

A valuation domain is an integral domain R such that for all a,b ϵ R either a I b or b I a. (Clearly a discrete valuation ring is a valuation domain.) A Prüfer domain is an integral domain in which every finitely generated ideal is invertible.

(a) The following are equivalent:

(i) R is a Prüfer domain;

(ii) For every prime ideal P in R, R_P is a valuation domain;

(iii) For every maximal ideal M in R, R_M is a valuation domain.

(b) A Prüfer domain is Dedekind if and only if it is Noetherian.

(c) If R is a Prüfer domain with quotient field K, then any domains such that R ⊂ S ⊂ K is Prüfer.