Why is the Endomorphism ring of a finite dimensional central simple algebra isomorphic to the matrix ring over the centre/underlying field having dimension , where is the dimension of the algebra over the field?
Why is the Endomorphism ring of a finite dimensional central simple algebra isomorphic to the matrix ring over the centre/underlying field having dimension , where is the dimension of the algebra over the field?
Thank you both. My second question is:
As you know, a finite dimensional simple algebra is isomorphic to a matrix ring of a division ring.
If the algebra is central, why is the division ring then central?
Is it because the center of the division ring has to be contained in the centre of the matrix ring? Thus it must be the underlying field. Doesn't hurt to double check!!!