Artinian affine $K$-algebra [duplicate]

Let $K$ be a field and $A$ an affine $K$-algebra. Show that $A$ has (Krull) dimension zero (is artinian) if and only if it is finite dimensional over $K$.

Finite dimensional (as a vector space) algebras are zero dimensional: Using quotients, it is enough to prove that every finite dimensional domain is a field. But this is standard: The multiplication map by a nonzero element is injective, hence also surjective.

If $A$ is a zero dimensional finitely generated $k$-algebra, then it is artinian, and we may even assume that $A$ is local, say with maximal ideal $\mathfrak{m}$. Now try to prove that each $\mathfrak{m}^n/\mathfrak{m}^{n+1}$ is finite-dimensional. Since $\mathfrak{m}$ is nilpotent, this shows that $A$ is finite dimensional (as a $k$-vector space).