Prime T-Ideals in Polynomial and Power Series Rings Over a Pseudo-Valuation Domain

Abstract

Let R be an m-dimensional pseudo-valuation domain with residue field k, let V be the associated valuation domain with residue field K, and let k 0 be the maximal separable extension of k in K. We compute the t-dimension of polynomial and power series rings over R. It is easy to see that t-dim R[x 1,…, x n ] = 2 if m = 1 and K is transcendental over k, but equals m otherwise, and that t-dim R[[x 1,…, x n ]] = ∞ if R is a nonSFT-ring. When R is an SFT-ring, we also show that: (1) t-dim R[[x]] = m; (2) t-dim R[[x 1,…, x n ]] = 2m − 1, if n ≥ 2, K has finite exponent over k 0, and [k 0: k] < ∞; (3) t-dim R[[x 1,…, x n ]] = 2m, otherwise.