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I have the following question:

Show that the center of a semisimple unital ring is a product of a finite number of fields.

At first, I thought of showing that the center of a semisimple field is also semisimple. Since the center is commutative, it's a product of finite fields due to the Wedderburn-Artin theorem.
While this result seemed trivial to me, I found some difficulty proving it, since the center is not a module over the ring. I've tried to find the result online and I couldn't find anything. I wonder if the statement is even true.
I would like to figure out if I've missed something with this question. Algebra is not my strong point, so it'll be helpful for me to figure out what I'm missing.

CauchyChaos
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1 Answers1

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  1. The center of a product of rings is the product of their centers (just analogous to this.)
  2. The center of a matrix ring is are the constant diagonal matrices over the center of the base ring.
  3. The center of a division ring is a field.

Together this proves the center of a semisimple ring is a finite product of fields.

rschwieb
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