VALUATIVE DOMAINS

Abstract

A (commutative integral) domain R is said to be valuative if, for each nonzero element u in the quotient field of R, at least one of R ⊆ R[u] and R ⊆ R[u-1] has no proper intermediate rings. Such domains are closely related to valuation domains. If R is a valuative domain, then R has at most three maximal ideals, and at most two if R is not integrally closed. Also, if R is valuative, the set of nonmaximal prime ideals of R is linearly ordered, at most one maximal ideal of R does not contain each nonmaximal prime of R, and RP is a valuation domain for each prime P except for at most one maximal ideal. Any integrally closed valuative domain is a Bézout domain. Valuation domains are characterized as the quasilocal integrally closed valuative domains. Each one-dimensional Prüfer domain with at most three maximal ideals is valuative.