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Consider the subalgebra $R=k[x,xy,xy^2,xy^3, \ldots] \subset k[x,y]$. How do I prove that $R$ is not finitely generated (over $k$)?

What is the general strategy for proving that an algebra is not finitely generated?

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$k$ is any field. – Mohan Jan 10 '13 at 8:08
Can a non-noetherian ring finitely generated over a field ? – user18119 Jan 10 '13 at 8:33
up vote 2 down vote accepted

Suppose that $R$ is generated by $f_1,\dots,f_n$ over $k$, which are polynomials in terms of $xy^m$ up to $m = M$. ($M$ is finite since there are finitely many $f_1,\dots,f_n$). This would imply that $R = k + (x,xy,\dots,xy^M)$. Now we can go back to your previous question to see that for example, $xy^{M+1}$ does not lie in $(x,xy,\dots,xy^M)$.

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This is a little strange to me $k[x]$ is generated by $x$ and $(x)$ is a proper ideal. – JSchlather Jan 10 '13 at 8:13
@jacobSchlather, thanks, it's corrected. – user27126 Jan 10 '13 at 8:15

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