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Recall from The Coordinate Ring of an Affine Variety page that if $K$ is a field and $V \subseteq \mathbb{A}^n(K)$ is a nonempty affine variety then the coordinate ring of $V$ is defined as:
(1)
\begin{align} \quad \Gamma(V) = K[x_1, x_2, …, x_n]/I(V) \end{align}
We noted that since $V$ is an affine variety, $I(V)$ is a prime ideal and so $\Gamma(V)$ is an integral domain.
Definition: Let $K$ be a field and let $V \subseteq \mathbb{A}^n(K)$ be a nonempty affine variety. The Set of Functions from $V$ to $K$ is denoted by $\mathcal F(V, K)$, and is a ring with the operation of addition defined for all $f, g \in \mathcal F(V, K)$ by $(f + g)(x) = f(x) + g(x)$ and the operation of multiplication defined for all $f, g \in \mathcal F(V, K)$ by $(fg)(x) = f(x)g(x)$. 
Definition: Let $K$ be a field and let $V \subseteq \mathbb{A}^n(K)$ be a nonempty affine variety. A function $f \in \mathcal F(V, K)$ is a Polynomial Function if there exists a polynomial $F \in K[x_1, x_2, …, x_n]$ such that $f(\mathbf{p}) = F(\mathbf{p})$ for every $\mathbf{p} \in V$. The Set of Polynomial Functions from $V$ to $K$ is denoted by $\mathcal P(V, K)$. 
Definition: Let $K$ be a field and let $V \subseteq \mathbb{A}^n(K)$ and $W \subseteq \mathbb{A}^m(K)$. A Polynomial Map if a function $\varphi : V \to W$ such that there exists polynomials $T_1, T_2, …, T_m \in K[x_1, x_2, …, x_n]$ such that $\varphi(\mathbf{p}) = (T_1(\mathbf{p}), T_2(\mathbf{p}), …, T_m(\mathbf{p})$ for every $\mathbf{p} \in V$. A polynomial map $\phi : V \to W$ is said to be an Isomorphism if there exists another polynomial map $\psi : W \to V$ such that $\psi \circ \phi = \mathrm{id}_V$ and $\phi \circ \psi = \mathrm{id}_W$. 
If $V \subseteq \mathbb{A}^n(K)$ and $W \subseteq \mathbb{A}^m(K)$ are affine varieties, then a natural homomorphism exists between the two.
Theorem 1: Let $K$ be a field and let $V \subseteq \mathbb{A}^n(K)$ and $W \subseteq \mathbb{A}^m(K)$ be affine varieties and let $\varphi : V \to W$ be a polynomial map. Then the function $\phi : \mathcal F(W, K) \to \mathcal F(V, K)$ defined for all functions $f \in \mathcal F(V, K)$ by $\phi(f) = f \circ \varphi$ is a ring homomorphism. 
 Proof: Let $f, g \in \mathcal F(W, K)$. Then:
(2)
\begin{align} \quad \phi(f + g) = (f + g) \circ \varphi = (f \circ \varphi) + (g \circ \varphi) = \phi(f) + \phi(g) \end{align}
(3)
\begin{align} \quad \phi(fg) = (fg) \circ \varphi = (f \circ \varphi) \cdot (g \circ \varphi) = \phi(f) \cdot \phi(g) \end{align}
 So $\phi : \mathcal F(W, K) \to \mathcal F(V, K)$ is a ring homomorphism. $\blacksquare$
The following theorem tells us that there is a onetoone correspondence between the set of polynomial maps from two nonempty varieties and the set of homomorphisms from the corresponding coordinate rings.
Theorem 2: Let $K$ be a field and let $V \subseteq \mathbb{A}^n(K)$ and $W \subseteq \mathbb{A}^m(K)$ be nonempty affine varieties. Then there is a onetoone correspondence between the set of polynomial maps $\varphi : V \to W$ and the set of ring homomorphisms $\phi : \Gamma(W) \to \Gamma(V)$. 
Corollary 3: Let $K$ be a field and let $V \subseteq \mathbb{A}^n(K)$ and $W \subseteq \mathbb{A}^m(K)$ be nonempty affine varieties. Then a polynomial map $\phi : V \to W$ is an isomorphism if and only if $\Gamma(W)$ and $\Gamma(V)$ are ring isomorphic. 
Related post:
 The Multiplicity of an Affine Plane Curve at a Point
 Simple and Multiple Points of Affine Plane Curves
 Affine Plane Curves
 Polynomial Forms
 Discrete Valuation Rings
 Local Rings
 The Rational Function Field of an Affine Variety
 The Coordinate Ring of an Affine Variety
 Affine Varieties
 Corollary to the Hilbert Basis Theorem