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This was the problem that was given: Let $k$ be a field and define $R = \frac{k[x, y]} {< x^2 >}$ Prove that R cannot be the coordinate ring of a variety $V \subseteq A^2_k$.

I'm confused about what the question is actually asking. I know that a coordinate ring of a variety is of the form: $\frac{k[x_1,x_2...x_n]}{I(V)}$. However, I don't really understand that $R$ cannot be a coordinate ring in this case.

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A coordinate ring is always a reduced ring, because $I(V)$ is necessarily radical. On the other hand $k[x,y]/(x^2)$ has some obvious nonzero nilpotent elements.

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  • $\begingroup$ Can you please clarify what you mean by a reduce ring and what you mean by nonzero nilpotent elements? $\endgroup$ – Masha Apr 30 at 23:27
  • $\begingroup$ The terminology is standard, both in English books an on English internet, as far as I know. $\endgroup$ – Saucy O'Path Apr 30 at 23:27
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    $\begingroup$ You're right - I double checked my notes and found it, thank you. So would I be correct in saying that in this case we have to prove that $R$ cannot be a reduced ring, and therefore must find a non-zero nilpotent element? In that case, don't we know it cannot be a reduced ring because if $x = 0$ then $x^2 = 0$ and therefore it is nilpotent? $\endgroup$ – Masha Apr 30 at 23:34
  • $\begingroup$ That was the point. $\endgroup$ – Saucy O'Path Apr 30 at 23:35
  • $\begingroup$ Thanks for the assistance! $\endgroup$ – Masha Apr 30 at 23:36

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