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I am trying to solve a problem that asks to classify the semisimple rings of order $5^4$. It is a question of a past exam and I am not really sure about the answer.

What I know; semisimple rings are finite direct products of matrix algebras over division rings. I can figure out that a finite division ring is a field (Wedderburn's theorem).

What I have is that $$R\simeq M_{n_1}(k_1)\times\dots\times M_{n_s}(k_s)$$ where $|R|=5^4$ and the $k_i$s are finite fields. Therefore, $5^4=| k_1|^{n_1}\cdots |k_s|^{n_s}$.

From there, should I just start taking cases or there is any other restriction?

user128787
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1 Answers1

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A finite division ring is commutative. The $n\times n$ matrix ring over a finite field has cardinality $|F|^{n^2}$, so you have just a few choices.

The base field can have up to $5^4$ elements and you can choose between $F_1=GF(5)$, $F_2=GF(5^2)$, $F_3=GF(5^3)$ and $F_4=GF(5^4)$ (where $GF(r)$ denotes the field with $r$ elements).

Now combine products of matrix rings over these fields to yield $5^4$ elements.

egreg
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