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Let A, B be two NON-isomorphic finitely generated k-algebras, is it possible they isomorphic as abstract commutative unitary rings? (any concrete examples?)

If the answer to the above is possibly yes, then what assumptions should one add to preserve the NON-isomorphicity?

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up vote 5 down vote accepted

Yes. For example, let $k$ be $\mathbb{Q}(x)$, let $A$ be $k$, and let $B$ be $k\left(\sqrt{x}\right)$.

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