As in title. Are there only finite many maximal left ideals for a left Artinian ring?
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1$\begingroup$ Could you please add some motivation? Also, see mathoverflow.net/faq#whatquestions $\endgroup$– David Roberts ♦Commented Feb 17, 2011 at 6:34
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1$\begingroup$ And please make the body of the question be complete, even if that means you have to repeat part of the title! $\endgroup$– Mariano Suárez-ÁlvarezCommented Feb 17, 2011 at 6:40
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3$\begingroup$ Have you considered this question for the ring $M_2(\mathbb C)$? $\endgroup$– EmertonCommented Feb 17, 2011 at 6:40
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$\begingroup$ (Note that the answer is yes in the commutative case: this is part of the standard structure theory for Artinian rings.) $\endgroup$– Pete L. ClarkCommented Feb 17, 2011 at 7:02
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2 Answers
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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?
answered Feb 17, 2011 at 6:43
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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 $A$.
