WHEN ARE THE PRIME IDEALS OF THE LOCALIZATION R[X](T) EXTENDED FROM R

Abstract

Let R be an integrally closed integral domain, {X-alpha} a set of indeterminates over R, and T a multiplicatively closed subset of R[{X-alpha}]. We prove the equivalence of the following statements: (1) Every prime ideal of R[{X-alpha}](T) is extended from R. (2) Every ideal of R[{X-alpha}](T) is extended from R. (3) Every principal ideal of R[{X-alpha}](T) is extended from R. (4) There exists a Prufer v-multiplication overring A of R such that R[{X-alpha}](T) = A(v), where A(v) is the Kronecker function ring of A with respect to the v-operation. The case when R is not integrally closed is also taken care of. Similar statements for rings with zero divisors are considered and their equivalence is established.