Isomorphism of polynomial rings implying isomorphism of the coefficient rings

Question

Let $R$ and $S$ be commutative rings. Let $x, y$ be indeterminates, and assume that one has an isomorphism $R[x] \rightarrow S[y]$ (not necessarily mapping $x$ to $y$ of course). Does this imply $R \cong S$? If not, what is a counterexample?

This may seem like a homework problem, but is not.

Alex Becker · Accepted Answer · 2013-05-10T09:20:53.993

3

This was settled in the negative by M. Hochster in 1972. His paper can be found here.

edited May 10 '13 at 09:20

answered May 10 '13 at 07:11

Alex Becker

60,569

Wonderful! Thanks! – Rankeya May 10 '13 at 07:14
settled in the sense: There are counterexamples. – Martin Brandenburg May 10 '13 at 09:00

Isomorphism of polynomial rings implying isomorphism of the coefficient rings

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