Question

Show that for any ring R, the center of the matrix ring Mn(R) consists of the diagonal matrices r • In, where r belongs to the center of R.

Answer 1

Let $E_{ij}$ be the matrix units. If $r \in Z(R)$ , then $(r \cdot I_n)(aE_{ij}) = raE_{ij} = (aE_{ij})(rI_n)$ , so $r \cdot I_n \in Z(S)$ , where $S = \mathbf{M}_n(R)$ . Conversely, consider $M = \sum r_{ij}E_{ij} \in Z(S)$ . From $ME_{kk} = E_{kk}M$ , we see easily that $M$ is a diagonal matrix. This and $ME_{kl} = E_{kl}M$ together imply that $r_{kk} = r_{ll}$ for all $k, l$ , so $M = r \cdot I_n$ for some $r \in R$ . Since this commutes with all $a \cdot I_n(a \in R)$ , we must have $r \in Z(R)$ .