As in title. Are there only finite many maximal left ideals for a left Artinian ring?

Consider the ring $M_2(k)$ of $2\times 2$ matrices over a field $k$, which is Artinian. Can you describe the maximal left ideals? 


As indicated by Pete Clark, an Artinian commutative ring has only finitely many maximal ideals. However, the correct generalization is not the one you seem to expect but the following: If $A$ is rightArtinian, the set of isomorphism classes of simple right$A$modules is finite. This implies the commutative case. Assume that $A$ is commutative. Then, if $S$ is a simple $A$module, there exists a maximal ideal $\mathfrak m$ of $A$ such that $S\simeq A/\mathfrak m$; moreover, $\mathfrak m$ is the annihilator of $S$ so that there is a bijection between isomorphism classes of simple $A$modules and maximal ideals of $A$. 

