Proving the scalar matrices are the center of the matrix ring

[Bipolarity]

I read that scalar matrices are the center of the ring of matrices. How would I prove this?
Tips are appreciated. It is already obvious that scalar matrices commute with all matrices, but the converse seems tricky.

BiP

 

[Axiomer]

Supposing R is commutative with identity:

Let C be in the center of M(R). If C is not diagonal then c_ij≠0 for some i≠j. Letting A be the matrix with 1 in the ji position and 0 elsewhere, we see that CA≠AC, contradicting the fact that C is in the center of M(R). So C is diagonal. If C is diagonal but c_ii≠c_jj, then we can again find a matrix in M(R) which does not commute with C.