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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!

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  • $\begingroup$ $M_n(\mathbb{F}_{p^k})$ is noncommutative iff $n>1$, to see this just find two matrices that don't commute. Writing $2^9$ as a product of numbers of the form $2^{kn^2}$ is the same as writing 9 as a sum of number of the form $kn^2$, there really aren't that many ways so it's probably easiest to just list them. $\endgroup$ – Nate Apr 26 '14 at 19:29
  • $\begingroup$ Groups of order $p^1$ and $p^2$ are abelian, so your direct product must have at least one factor $p^{\geq 2}$? This seems to require $2^{1\cdot 3^2}$, $2^{2 \cdot 2^2}$, $2^{1 \cdot 2^2}$, or $2^{\{3,4,5,6,7,8,9\} \cdot 1^2}$ for this "large factor"? $\endgroup$ – Eric Towers Apr 26 '14 at 19:32
  • $\begingroup$ Are you sure the ring has to be not commutative? Most sources use the adjective t mean "not necessarily commutative (but could be)" $\endgroup$ – rschwieb Apr 26 '14 at 22:05
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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.

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