A note on the uniqueness of rings of coefficients in polynomial rings

Abstract:

We say that the ring A is of transcendence degree n over its subfield k if for every prime $P \subset A$ the transcendence degree of $A/P$ over k is at most n and equality is attained for some P. In this paper we prove the following: Suppose A is a noetherian ring of transcendence degree one over its subfield k. Then if B is any ring such that the polynomial rings \[ A[{X_1}, \cdots ,{X_m}]\quad {\text {and}}\quad B[{Y_1}, \cdots ,{Y_m}]\] are isomorphic, A is isomorphic to B. Moreover if A has no nontrivial idempotents then either A is isomorphic to the polynomials in one variable over a local artinian ring or, modulo the nil radical, the given isomorphism takes A onto B.