I managed to prove that if k[X] and k[Y] are isomorphic as k-algebras, then X is isomorphic to Y as affine algebraic sets.
My question is however, does this guarantee that X and Y lie within the same affine space i.e. is the dimension the same? When I worked out this result, dimension never seemed to come into it so I was a bit worried I did something wrong...
Thanks for any help