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

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 of prime ideals of R can be embedded in Spec(D), i.e., can be covered by some isomorphic tree 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 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 , what are necessary and sufficient conditions that be a tree in Spec(A)? We also prove that if R is an integral domain with the following property: for a given tree in Spec(R), there exists a PrüferoverringP(R) of R with the tree such that and , then an integral and mated extension of R has the same property.