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I have seen statements of a special case of Wedderburn's Structure Theorem:

If G is a finite group, then $\mathbb{C}G\cong R_1\bigoplus R_2\bigoplus \cdots \bigoplus R_k$, for some $k$, where each $R_i\cong M_{n_i}(\mathbb{C})$ for some $n_i$. (We must have $\sum_{i=1}^k n_i^2=|G|$.)

Is there a proof of this using ring theory, without using modules, algebras, etc...

Thanks a lot!

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    $\begingroup$ Wait, I thought ring theory was modules, algebras, etc. :) $\endgroup$ – Thomas Andrews Oct 18 '15 at 5:11
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    $\begingroup$ This implies and so is at least as hard as Maschke's theorem, and once you have Maschke's theorem you're pretty close to this once you know some fairly straightforward facts about modules. I really do not recommend avoiding learning about modules. This is a very fundamental part of ring theory. $\endgroup$ – Qiaochu Yuan Oct 18 '15 at 5:18
  • $\begingroup$ @Thomas Andrews Sorry, you are probably right... I haven't learned about modules or algebras yet. Didn't know they were part of ring theory! I guess I am just looking for a proof that doesn't involve modules or algebras... $\endgroup$ – Esmath Oct 18 '15 at 5:18

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