From: wcw@math.psu.edu (William C Waterhouse) Subject: Re: polynomial rings Date: 18 Nov 1999 22:28:58 GMT Newsgroups: sci.math Keywords: isomorphic polynomial rings have the same dimension In article <80rs5r\$23d\$1@nnrp1.deja.com>, Robin Chapman writes: >... > > R[x,y] is not isomorphic to R[x,y,z]. > > No. THis is done in elementary texts on algebraic geometry and/or > commutative algebra. The ring R[x_1,x_2, ..., x_n] (I presume R > is the real field) has a chain of n+1 prime ideals > 0 <= <= <= ... <= > but no longer chain (this is not obvious). Hence n is determined > by the ring structure of R[x_1, ..., x_n]. Here's a proof that doesn't use any results from dimension theory. Theorem Let F and F' be fields. Then the polynomial rings F[X,Y] and F'[U,V,W] cannot be isomorphic as rings. Proof. Suppose we have an isomorphism h from F[X,Y] onto F'[U,V,W]. It must take invertible elements to invertible elements, so it induces an isomorphism of F onto F'. We may thus replace F' by F and assume that the isomorphism is the identity on constants. Look now at the F-homomorphism k:F[X,Y] -> F sending X and Y to 0. Then composition with the inverse of h gives an F-homomorphism from F[U,V,W] to F. It must send U,V,W to constants a,b,c. There is an automorphism of F[U,V,W] taking U-a to U, etc, so we can change our choice of variables in the second ring and assume that the composite sends U,V,W to 0. Now the kernel of k is the ideal M generated by X and Y. Our h must take this to the ideal M' generated by U,V,W. But then h must also take M^2 to (M')^2. This however is impossible, since M/M^2 is an F-space of dimension 2 and M'/(M')^2 has dimension 3. Corollary. Let R be a (commutative) ring. Then there is no ** R-algebra** isomorphism from R[X,Y] onto R[U,V,W]. Proof. Let N be a maximal ideal in R, with R/N = F (a field). Then any R-algebra isomorphism from R[X,Y] onto R[U,V,W] will induce an isomorphism from F[X,Y] onto F[U,V,W]. Remark. There can sometimes be ring isomorphisms, as for example when R itself is already a polynomial ring in infinitely many variables. William C. Waterhouse Penn State