ON CONJUGATE PRIME IDEALS OF TENSOR PRODUCTS OF k-ALGEBRAS

Abstract

The purpose of this paper is to explore new aspects of the prime ideal structure of tensor products of algebras over a field k. We prove that given a k-algebra A and a normal field extension K of k (in the sense of Galois theory), then for any prime ideals P1 and P2 of K ⊗k A lying over a fixed prime ideal p of A, there exists a k-automorphism σ of K such that (σ ⊗k id A)(P1) = P2. As an Application, we establish a result related to the dimension theory of tensor products stating that, for two arbitrary k-algebras A and B, the minimal prime ideals of p ⊗k B + Aσk q have the same height, for any prime ideals p and q of A and B, respectively.