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Let $\mathcal{M}_1,\dots,\mathcal{M}_r$ be all the maximal ideals of an Artin ring $A$ which is a finite $\mathbb{K}$-algebra; so let $A/\mathcal{M}_1\cdots\mathcal{M}_r$ be a $\mathbb{K}$-vector space. Which is its dimension?

Is it a general case that the number of maximal ideals in $A$ is equal to $\operatorname{dim}_{\mathbb{K}} A+1$?

user26857
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TheWanderer
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  • I've found a post, titled "A finitely dimensional algebra over a field has only finitely many prime ideals all of them are maximal" which gives the answer that I'm searching for. – TheWanderer Feb 12 '15 at 11:25

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