Proving the scalar matrices are the center of the matrix ring

  • Thread starter Bipolarity
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Main Question or Discussion Point

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
 

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  • #2
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Supposing R is commutative with identity:

Let C be in the center of M(R). If C is not diagonal then cij≠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 cii≠cjj, then we can again find a matrix in M(R) which does not commute with C.
 

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