# Classification of all semisimple rings of a certain order [closed]

I'd appreciate it if you tell me where to begin in order to solve this question:

Classify (up to ring isomorphism) all semisimple rings of order 720.

Could the Wedderburn-Artin Structural Theorem be applicable?

• Since you're new, I'd like to give you some advice about the site: To get the best possible answers, you should explain what your thoughts on the problem are. That way, people won't tell you stuff you already know, and they can write answers at an appropriate level; also, people tend to be more willing to help if you show that you've tried the problem yourself. Lastly, some may consider your post rude because it is phrased as a command, not a request for help, so please consider rewriting it. Aug 11 '13 at 5:11
• We clearly need rschwieb here. :) Aug 11 '13 at 5:28
• I'm surprised that this question got 2 downvotes and 3 close votes. What's wrong with it? Aug 11 '13 at 6:34
• I opend a meta question about this. meta.math.stackexchange.com/questions/10580/… Aug 11 '13 at 6:41
• This is a borderline case, but I think the question shouldn't be closed now. The subsequent edits now provide (a modicum) of their thoughts by asking about applying the Artin-Wedderburn theorem, and they're only asking for a hitn now. This question is easily hintable, after all. Aug 11 '13 at 17:49

## 1 Answer

Yes, you should definitely apply Artin-Wedderburn.

The thing you gain from knowing the ring is finite is that the ring will be a product of matrix rings over fields, since finite division rings are fields. Hopefully you know that all finite fields are of prime power order.

Now then, an n by n matrix ring over a field with q elements clearly has $q^{n^2}$ matrices. Start deducing what the possibilities are :)