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I was wondering if the following statements were true;

1) Every semi-simple ring is a product of simple rings.

2) Every module over a division ring $R$ is free.

I think both of these statements are false but cannot come up with any counterexamples. Does anyone have any ideas?

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Both statements are true. The first follows from Artin-Wedderburn and the proof of the second is the same as the corresponding proof for fields (Zorn's lemma). – Qiaochu Yuan May 24 '12 at 20:30
up vote 3 down vote accepted

If you use "semisimple" as I do (to mean "$R$ is the direct sum of its simple right ideals" or "$R$ is Artinian with $rad(R)=0$") then the Artin-Wedderburn theorem proves the first statement true.

For the second question, I assume you are familiar with the proof that every vector space over a field $F$ has a basis. Once you know that is true, and $V$ has a basis $\{v_i\mid i\in I\}$, then you can map elements of $V$ to their coefficients in $\bigoplus_{i\in I} F$ to produce an isomorphism, showing that $V$ is free. If you review try this with division rings, you will find that commutativity was not necessary, and everything goes through here as well.

You also can try to work out the converse: if all right $R$ modules are free, then $R$ is a division ring.

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