# Proving that something cannot be a coordinate ring of an affine variety

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.
• 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? – Masha Apr 30 at 23:34