On the uniqueness of the coefficient ring in a polynomial ring

Abstract

If k is a field and X and Y are indeterminates then the statement “consider R = Iz[X, Y] as a polynomial ring in one variable” is ambiguous, for there arc infinitely many possible choices for the ring of coefficients (e.g., If A, = k[X f Y”] then A,[Y] == B,,[Y] 7: R but A,, f J,, if m # n). On the other hand, if Z denotes the integers then the polynomial ring Z[X] has a unique subring over which it is a polynomial ring. This investigation began with our consideration of the first of these examples. In fact, Coleman had asked: If k is a field, then although k[X, 1’1 can be written as a polynomial ring in many different ways, is it true that all of the possible coefficient rings are isomorphic? That is, if T is transcendental over d and 4[T] == k[X, Y], is A a polynomial ring over k ? We found that this is indeed the case (see our (2.8)).We next proved the following: If il is a one dimensional afine domain over a$eZd and B is a ring such that A [-Xl = B[ E;] zs an equality of polynomial rings, then either A ~ B or there is afield k such that each of A and B is a polynomial +zg in one variabZe over k. This is a corollary of (3.3) in the present paper. Our (7.7) sketches a version of the original proof. In studying this argument, we found that there were implicit in it techniques for investigating the following general question: Suppose A and B are commutative rings with identity and the polynomial rings 4 [X, , . . . , X,,] and B[E; ,..., I’,] are isomorphic, how are A and B related? Are A and B isomorphic? In particular, when does the given isomorphism take i4 onto B ? This study is mainly centered on the latter portion of the question. We are concerned almost entirely with domains It is convenient to use the following terminology which is modeled after that