AGREEABLE DOMAINS

An integral domain D with quotient field K is defined to be agreeable if for each fractional ideal F of D[X] with F subset of or equal to K[ X] there exists 0 not equal s is an element of D with sF subset of or equal to D[X]. D is agreeable double left right arrow D satisfies prop erty () (for 0 not equal f(X) is an element of K[X], there exists 0 n ot equal s is an element of D so that f(X)g(X) is an element of D[X] f or g(X) is an element of K[X] implies that sg(X) is an element of D[X] ) double left right arrow D[X] is an almost principal domain, i.e., fo r each nonzero ideal I of D[X] with IK[X] not equal K[X], there exists f(X) is an element of I and 0 not equal s is an element of D with sI subset of or equal to (f(X)). If D is Noetherian or integrally closed, then D is agreeable. A number of other characterizations of agreeable domains are given as are a number of stability properties. For exampl e, if D is agreeable, so is [GRAPHICS] and for a pair of domains D sub set of or equal to D' with [D : D'] not equal 0, D is agreeable double left right arrow D' is agreeable. Results on agreeable domains are us ed to give an alternative treatment of Querre's characterization of di visorial ideals in integrally closed polynomial rings. Finally, the va rious characterizations of D being agreeable are considered for polyno mial rings in several variables.