Rings of polynomials

Abstract

For an algebra R over a field k, with residue field K to be a ring of polynomials in one variable over k it is necessary that tr deg K/k = 1. We prove that under the hypothesis tr deg K/k = 1, R is a ring of Krull-dimension at most one. This is used to derive sufficient conditions for R to be a ring of polynomials in one variable over k.