All the semisimple rings with center a field

Question

Hi there I am trying to find every Semisimple ring that has center a field. I started with Weddeburn’s Theorem and since R is semisimple every matrix ring will be a field. So the center will be a product of fields is my thought correct? But then I don’t know how to continue any ideas?

Also I need to prove that the center of the group algebra of S3 has dimension 3 is it related to the previous question?

Answer 1

I think by every matrix ring will be a field. you mean the center of each matrix ring over a division ring is a field which is true.

And yes, your idea that if the ring is semisimple, the center would be the product of the centers of its factors. So yes, a semisimple ring whose center is a field is necessarily a simple ring already.

Also I need to prove that the center of the group algebra of S3 has dimension 3

Depending on the field, the group algebra will be semisimple, so yes it could be applied. But unfortunately you have to pick the base field to get an answer.