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.