the task is to prove that center of a matrix ring is the scalar matrices .
I found two more general proofs:
Center of an Algebra « Abstract Nonsense
The center of a matrix ring over a commutative ring is precisely the scalar matrices « Project Crazy Project
However I didn't understand those totally, so I coudn't reduce them to this simpler case. But I think the main point is:
Let's assume that and .
Matrix multiplication is commutative ( ) iff is in form .
I don't know how to continue etc., so any help is appreciated. Thanks!