Let $k$ be an [algebraically closed field](/page/Algebraically%20Closed%20Field), and let $X$ and $Y$ be affine varieties over $k$. Let $\iota:X\to Y$ be a regular map. Then $\iota$ is a closed embedding if and only if the pullback homomorphism $\iota^*:k[Y]\to k[X]$ is surjective.