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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0894J980RK]].\nThe *center* of $R$ is the set\n$$Z(R) = \\set{z \\in R : zr = rz \\text{ for all } r \\in R}.$$\n\n## Remarks\n\n$Z(R)$ is a commutative [[subring|0859D58WPD]] of $R$, and $R$ is commutative if and only if $Z(R) = R$.\nExample: for a field $K$ and $n \\ge 1$, the center of $M_n(K)$ is the set of scalar matrices $cI_n$ with $c \\in K$.\nThe 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.\nNon-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 $E_{12} E_{11} = 0$.\nSome authors write $C(R)$ for the center.\n",
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  "name": "Center of a ring",
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