Prime ideals of finite height in polynomial rings

We investigate the structure of prime ideals of finite height in polynomial extension rings of a commutative unitary ring R. We consider the question of finite generation of such prime ideals. The valuative dimension of prime ideals of R plays an important role in our considerations. If X is an infinite set of indeterminates over R, we prove that every prime ideal of R[X] of finite height is finitely generated if and only if each P is an element of Spec(R) of finite valuative dimension is finitely generated and for each such P every finitely generated extension domain of R/P is finitely presented. We prove that an integrally closed domain D with the property that every prime ideal of finite height of D[X] is finitely generated is a Prufer v-multiplication domain, and that if D also satisfies d.c.c. on prime ideals, then D is a Krull domain in which each height-one prime ideal is finitely generated.