Agreeable domains

An integral domain D with quotient field K is defined to be agreeable if for each fractional ideal F of DX] with F C KX] there exists 0 = s ε D with sF C DX]. D is agreeable ? D satisfies property (*) (for 0 ^ f(X) G KX], there exists 0 = s ε D so that f(X)g(X) ε DX] for g(X) ε KX] implies that sg(X) ε DX]) &; DX] is an almost principal domain, i.e., for each nonzero ideal I of DX] with IKX] = KX], there exists f(X) ε I and 0 = s ε D with sI C (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 example, if D is agreeable, so is ?^αD_{P α} and for a pair of domains D?D′ with a DD:′]≠0, D is agreeable?D′ is agreeable. Results on agreeable domains are used to give an alternative treatment of Querre's characterization of divisorial ideals in integrally closed polynomial rings. Finally, the various characterizations of D being agreeable are considered for polynomial rings in several variables.