Lifting up a tree of prime ideals to a going-up extension

Abstract

We prove that if R⊆ D is an extension of commutative rings with identity and the going-up property (for example, an integral extension), then any tree T of prime ideals of R can be embedded in Spec( D), i.e., T can be covered by some isomorphic tree T′ of prime ideals of D. In particular, the prime spectrum of a Prüfer 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 Prüfer domain. In this case, we obtain two Prüfer domains such that Spec( R)↪ Spec( D). We also prove that for an integral domain R, there exists a Bézout domain D such that any tree T⊆Spec(R) can be embedded in Spec( D). We give a sufficient condition for the question: given an extension A⊆ B of commutative rings and a tree T⊆Spec(B) , what are necessary and sufficient conditions that T c={Q∩A|Q∈ T} be a tree in Spec( A)? We also prove that if R is an integral domain with the following property: for a given tree T in Spec( R), there exists a Prüfer overring P( R) of R with the tree T′ such that ( T′) c= T and T≅ T′ , then an integral and mated extension of R has the same property.