Rings of polynomials

Abstract

For aii algebra R over a field k, with residue field K to be a ring of polyniomials in one variable over k it is necessary that trdeg K/k = 1. We prove that under the hypothesis tr* deg K/k -1, R is a ring of Krtull-dimension at most one. This is used to derive sufficient conditions for R to be a ring of polynomials in one variable over k. 1. Let k be a subfield of the commutative ring R. Let K be the quotient field of R. The problem we are concerned witlh is: When is R a ring of polynomials? IIn a previous paper [1] we obtained the following result: If R is a subring of k [Xi . . . Xn] such that with every element of R all of its factors in k [XIc xn] already lie in R, and if tr deg K/k =n, then R is a ring of polynomials. One of the results that we get in this paper is that R is a ring of polynomials also in case tr deg K/k = 1. We start by studying rings R for wlhich tr deg K/k_? 1. We prove that if R is a unique factorization domain, and R is a subring of the ring of polynomials k [xI . . . Xn], then R is a ring of polynomials. For subrings of the rings of polynomials over k we prove that (i) if R is a principal ideal domain then R is a ring of polynomials, and (ii) if R has a principal ideal M so that R/M is canonically isomorphic to k, then R is a ring of polynomials. Some possible generalizations and modifications are also pointed Dut. 2. The main object of this section is the study of the rings R for which tr deg K/k? 1. THEOREM I. If kCR, and if tr deg K/k ?1, then Krull-dim R 1, and let us lerive a contradiction. Since there exist prime ideals P, Q in R so that Received by the editors November 12, 1969. A IS Subject Classifications. Primary 1393; Secondary 1230, 1420. Key WVords antd Phrases. Rings of polynomials, rings of power series, unique facorization domain, principal ideal domain, Euclidean domain, transcendence degree, Irull dimension.