# Centre of a matrix ring are diagonal matrices

Show that $Z(M_n(R))$ consist of diagonal matrices with entries $Z(R)$.

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Suppose $A=(a_{ij})\in Z(M_n(R))$. Let $E_{ij}$ be the matrix whose $i,j$ entry is $1$, and all other entries are $0$. Then the equations $$E_{ii}A=AE_{ii}$$ for $1\leq i\leq n$ implies that $A$ is necessarily diagonal. (Why?) Furthermore, $$AE_{ij}=E_{ij}A$$ for $1\leq i,j\leq n$ implies that $a_{ii}=a_{jj}$ for all $i$ and $j$. (Why?) Hence $A=aI_n$ for some $a\in R$. But notice that $$aI_n(bI_n)=bI_n(aI_n),\quad \forall b\in R$$ implies that $a\in Z(R)$.