As in title. Are there only finite many maximal left ideals for a left Artinian ring?
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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? |
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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 right-Artinian, 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 $A$. |
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