Let R be a commutative
ring with identity. R is said to have dimension n, written dimR = n, if there exists a
chain P0⊂ P1⊂⋯⊂ Pn of n + 1 prime ideals of R, where Pn⊂ R, but no such
chain of n + 2 prime ideals. Seidenberg has shown that if dimR = n and X is an
indeterminate over R, then n + 1 ≦dimR[X] ≦ 2n + 1. Moreover, he has shown that
dimR[X] = n + 1 if R is a Prüfer domain. The author has shown that if V is a rank
one nondiscrete valuation ring, then dimV [[X]] = ∞. The principal result of
this paper is that if D is a Prüfer domain with dimD = n, then either
dimD[[X]] = n + 1 or dimD[[X]] = ∞, and necessary and sufficient conditions are
given.
