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Let $F$ be an algebraically closed field and let $R$ be a finite-dimensional $F$-algebra and suppose that $R$ has no nonzero nil ideals. Show that $R$ is isomorphic to a finite direct product of matrix rings over $F$.

I think it might be useful to use the Artin-Wedderburn theorem. I know that if $R$ is a finite-dimensional $F$-algebra then $R$ is an artinian ring, but I'm not sure how to prove this.

However, if I can prove that claim, by the Artin-Wedderburn theorem, $R$ is isomorphic to a direct product of matrix rings over division rings, but these division rings might not be $F$, so I was wondering how I could fix this?

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For your first question, a right ideal is also an $F$ subspace. A proper containment of subspaces must experience a drop of at least one dimension. This cannot continue indefinitely.

For your second question, you should probably double check your version of the A-W theorem. Usually for algebras it’s stated as a product of matrix rings “…over division rings that are finite dimensional extensions of $F$”. It’s not hard to show an algebraically closed field cannot have (proper) extensions like that.

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