PRIME T-IDEALS IN POLYNOMIAL AND POWER SERIES RINGS OVER A PSEUDO-VALUATION DOMAIN

摘要

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[x1, ... , xn] = 2 if m =1 and K is transcendental over k, but equals m otherwise, and that t-dim R[x(1), ... , xn] = infinity 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] < infinity; (3) t-dim R[x(1), ... , x(n)] = 2m, otherwise.