Let $k$ be an algebraically closed field.

(a) Let $X$ and $Y$ be varieties defined over $k$. Given a function $f: X \rightarrow Y$, define what it means for $f$ to be a morphism of varieties.

(b) If $X$ is an affine variety, show that the coordinate ring $A(X)$ coincides with the ring of regular functions on $X$. [Hint: You may assume a form of the Hilbert Nullstellensatz.]

(c) Now suppose $X$ and $Y$ are affine varieties. Show that if $X$ and $Y$ are isomorphic, then there is an isomorphism of $k$-algebras $A(X) \cong A(Y)$.

(d) Show that $Z\left(x^{2}-y^{3}\right) \subseteq \mathbb{A}^{2}$ is not isomorphic to $\mathbb{A}^{1}$.