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Prove that $X$ and $Y$ are isomorphic varieties if and only if $R[X]$ and $R[Y]$ are isomorphic rings. Where $R[X]$ and $R[Y]$ are rings of algebraic functions associated with $X$ and $Y$

If we have two maps $f : X \rightarrow Y$ and $g : Y \rightarrow X$
$X$ is isomorphic to $Y$ if $f \circ g = 1_Y$ and $g \circ f = 1_X$
The pullback maps are $f^* : R[Y] \rightarrow R[X]$ and $g^* : R[X] \rightarrow R[Y]$
How do I go further from here?

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    What is $R$ ? Don't you need $R$-algebras isomorphism? – 2017-01-29
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    @Watson R is the is the ring of functions associated with X and Y – 2017-01-29
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    So you mean over $\Bbb C$ or over another field? – 2017-01-29
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    @Watson over $C$ – 2017-01-29
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    If $R$ is the set of regular functions, then this is false, since for any projective variety $X$ over $\mathbb{C}$, $R[X]=\mathbb{C}$. Same applies if $R[X]$ is the field of meromophic functions, since this only implies birationality. So, the question is unclear. – 2017-01-29
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    @Mohan I suspect the OP means $X$ and $Y$ are affine varieties. – 2017-01-30

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