Let $A$ be an Artin ring that is also a finitely generated $K$-algebra. In particular, the krull dimension of $A$ is $0$. By Noether's Normalisation Lemma we have that $A$ is a $K$-vector space of finite dimension $\dim_{K}A$.
On the other hand, it is known that if $A$ is an Artin ring, $\mathrm{Spec}(A)$ is finite and discrete.
I was wondering how $\dim_{K}A$ and $\#\mathrm{Spec}(A)$ are related (If there is any relation). For example, does $\#\mathrm{Spec}(A)\leq \dim_{K}A$ hold?
Any help would be appreciated.
Since $A$ is finitely generated over a field (and commutative!), its nilradical coincides with its Jacobson radical $J$. It follows that $A$ and $A/J$ have the same spectrum. The latter is a commutative semisimple $K$-algebra, so Wedderburn's theorem tells us that it is a direct product of fields, each an extension of $K$, and its spectrum has one point per factor in this product.
It follows that the number of points in the spectrum is at most the dimension of $A/J$, which itself is at most the dimension of $A$.