LIFTING UP A TREE OF PRIME IDEALS TO A GOING-UP EXTENSION

Abstract

We prove that if R subset of or equal to D is an extension of commutative rings with identity and the going-up property (for example, an integral extension), then any tree F of prime ideals of R can be embedded in Spec(D), i.e., F can be covered by some isomorphic tree F of prime ideals of D. In particular, the prime spectrum of a Prufer domain can always be embedded in the prime spectrum of its integral extension. The most interesting case is when the integral extension is also a Prufer domain. In this case, we obtain two Prufer domains such that Spec(R) hooked right arrow Spec(D). We also prove that for an integral domain R, there exists a Bezout domain D such that any tree, F subset of or equal to Spec(R) can be embedded in Spec(D). We give a sufficient condition for the question: given an extension A subset of or equal to B of commutative rings and a tree F subset of or equal to Spec(B), what are necessary and sufficient conditions that F-c = {Q boolean AND A\Q is an element of } be a tree in Spec(A)? We also prove that if R is an integral domain with the following property: for a given tree F in Spec(R), there exists a Prufer overring P(R) of R with the tree F such that (F')(c) = F and F congruent to F', then an integral and mated extension of R has the same property. (C) 2003 Elsevier Science B.V. All rights reserved.