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.


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.

  • $\begingroup$ 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
  • 2
    $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.