Center of a ring

Remarks

$Z(R)$ is a commutative subring of $R$, and $R$ is commutative if and only if $Z(R) = R$. Example: for a field $K$ and $n \ge 1$, the center of $M_n(K)$ is the set of scalar matrices $cI_n$ with $c \in K$. The check: a matrix $Z$ that commutes with every matrix unit $E_{ij}$ satisfies $Z E_{ij} = E_{ij} Z$; comparing entries shows that $Z$ is diagonal with equal diagonal entries. Non-example: for $n \ge 2$, the matrix $E_{11}$ is not in the center of $M_n(K)$, since $E_{11} E_{12} = E_{12}$ but $E_{12} E_{11} = 0$. Some authors write $C(R)$ for the center.