Problem: Determine up to isomorphism all semisimple noncommutative rings of order 512 = $2^9$. (This is problem from an old qualifying exam I am studying from)
So far I have: Let A be a semisimple ring of order 512. A is finite so A must be Artinian. Then Wedderburn-Artin implies A is isomorphic to a direct product of matrix rings over division rings i.e. a direct product of $M_n(\mathbb{F}_{p^k})$ where $\mathbb{F}_{p_k}$ is a finite field of order $p^k$. Now the order of $M_n(\mathbb{F}_{p^k})$ is $p^{kn^2}$. So I'm assuming this becomes a combinatorics problem on how many ways to write $2^9$ as a product of numbers of the form $p^{kn^2}$. When I try to do this I get a very long list: $2^{9*1^2}, 2^{1*3^2}, 2^{8*1^2}*2^{1*1^2}, 2^{2*2^2}*2^{1*1^2}$ and so on. Is this the correct approach? Also, how do I know when A is noncommutative?
Thank you for your help!
A semisimple ring isn't commutative exactly when it has a nontrivial (I mean, dimension more than 1) matrix ring in its factorization.
Your "very long list" of four items seems to have exhausted the possibilities for this. The fields in question could only have order a power of 2. The possibilities include:
one or two 2 by 2 matrix rings over $F_2$
one 2 by 2 matrix ring over $F_4$
one 3 by 3 matrix ring over $F_2$
To count the commutative ones, you are just looking at partitions of 9 to determine the size of the fields to use in the factorization. There are 30 partitions.