ON THE KRULL DIMENSION OF TRANSCENDENTAL EXTENSIONS OF COMMUTATIVE RINGS

Abstract

If R is a commutative ring with identity and t ₁, t ₂,…, t _n, are transcendental over R such that no t_i is algebraic over the ring R[t ₁,…, t ₁-₁, t _i+1,…, t _n] it is proved that there exists a minimal prime ideal P in R such that dim R+1 ≤ dim R[t ₁,…, t _n] = dim R[t ₁,…, t _n]/P[t ₁,…, t _n]. This equality is used to prove results for R[t ₁,… t _n similar to those holding for the integral domain (R/P) [t ₁,…, t _n] with 31,… t ₁,…, t _n indeterminates over R/P.