 Solution

An isomorphism $\varphi :X\to Y$ of affine varieties and its inverse $\psi$ pull regular functions back to regular functions. By the preceding identification this gives k-algebra homomorphismsThey preserve sums, products, and every constant in $k$. Contravariance of pullback and $\psi \circ \varphi ={\mathrm{id}}_X$, $\varphi \circ \psi ={\mathrm{id}}_Y$ give ${\varphi }^{\ast }{\psi }^{\ast }={\mathrm{id}}_{A(X)}$ and ${\psi }^{\ast }{\varphi }^{\ast }={\mathrm{id}}_{A(Y)}$. Hence