If R R is a commutative integral domain with quotient field K K and x 1 , … , x n {x_1}, \ldots ,{x_n} are indeterminates, then there exist θ 1 , … , θ n {\theta _1}, \ldots ,{\theta _n} in K K such that dim ⁡ R [ x 1 , … , x n ] = n + dim ⁡ R [ θ 1 , … , θ n ] \dim R[{x_1}, \ldots ,{x_n}] = n + \dim R[{\theta _1}, \ldots ,{\theta _n}] .

Let R be a commutative ring with identity, A = R[[X]] and B = R[[Y]] with X and Y finite sets of indeterminates.Consider A and B as topological rings with the respective X and Y-adic topologies.If σ: A -• B is any R -homomorphism then there are R -automorphisms s and ί of Λ and B respectively, so that t°σ°s: A ->B is continuous.As a corollary we see that an R -endomorphism of A is surjective only if it is an automorphism.

Let Z Z denote the integers, Q Q the rationals, X X an indeterminate and G G a finitely generated abelian group. Then there is a Dedekind domain D D such that Z [ X ] ⊂ D ⫋ Q [ X ] Z[X] \subset D \varsubsetneqq Q[X] , and D D has class group G G . If G = 0 G = 0 then D D is a noneuclidian PID.

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

We say that the ring A is of transcendence degree n over its subfield k if for every prime P ⊂ A P \subset A the transcendence degree of A / P 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 \[ A [ X 1 , ⋯ , X m ] and B [ Y 1 , ⋯ , Y m ] A[{X_1}, \cdots ,{X_m}]\quad {\text {and}}\quad B[{Y_1}, \cdots ,{Y_m}] \] 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.

Let k be a field and {Xx}xe a a family of indeterminates over k.We show that if A is a ring of Knill dimension d such that k c A c. k[{Ax}xe^] then there are elements Yt,''■' •, Y* which are algebraically independent over k and a fc-isomorphism such that k c. (f>(A) c k[Ylt-■ ■ , Yd].This is used to show that a onedimensional ring A which satisfies the above conditions is necessarily an affine ring over k and is necessarily a polynomial ring if it is normal.In addition we show that such a ring A is a normal affine ring of transcendence degree two over k if and only if it is a two-dimensional Krull ring such that each essential valuation of A has residue field transcendental over k.