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From Lam's A First Course in Noncommutative Rings, section 1.3.

Let $R$ be a domain (EA: that is, a ring without zero divisors) such that $M_n(R)$ is semisimple. Show that $R$ is a division ring.

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My Answer: Break up $M_n(R)$ into left ideals $J_i = \{ \text{All columns but $i $ are zero}\}$. Now I claim without proof that left subideals of $J_i$ are all of the form $J_i \cap M_n(I)$, for $I \triangleleft R$ a left ideal. I further claim that $I$ is minimal iff $J_i \cap M_n(I)$ is minimal. Therefore to decompose $J_i$ as the sum of minimal left ideals is to decompose $R$ as the sum of minimal left ideals. So $R$ is semisimple.

Since $R$ is semisimple, $R = I_1 \oplus \dotsb \oplus I_n$, for $I_j$ minimal ideals. I claim without proof that the only way this can be a domain is if there is only one summand, that is, if $R$ has no left ideals, and so is a division ring.

I'm skeptical of my answer, because it would seem they would add as an intermediate step to prove that $R$ is semisimple. Do you see any problems? Or perhaps a more elegant way?

Eric Auld
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At a high level, one could think this way: $R$ is right Artinian iff $M_n(R)$ is right Artinian for every natural number $n$. If $M_n(R)$ is semisimple, then $R$ is an Artinian domain, and that is a division ring."

The problem is, of course, to show that the Artinian condition behaves that way. If you evaluate that proving "$M_n(R)$ right Artinian implies $R$ right Artinian" is not too much of a hardship, it would be a good way to go.

There are definitely good posts here on math.SE which can help you do this, for example:

https://math.stackexchange.com/a/26830/29335

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rschwieb
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  • I like this idea. It's cleaner. Would you say that my method is sound, albeit lengthy? – Eric Auld Mar 17 '16 at 00:15
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    @EricAuld To be frank, I think that a seven sentence proof is not really very lengthy, nor very strong when three statements are claimed without proof, some not being totally obvious. Perhaps you can still attempt to add finer trained detail without getting too lengthy: maybe you will see a more elegant explanation when you work out the details – rschwieb Mar 17 '16 at 03:11