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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0894J980RK]].\nThe ring $R$ is *noncommutative* if there are elements $a, b \\in R$ with $ab \\ne ba$.\n\n## Remarks\n\nExample: the ring $M_2(\\R)$ of $2 \\times 2$ real matrices is noncommutative.\nWith $E_{12}$ the matrix with $1$ in position $(1,2)$ and $0$ elsewhere, and $E_{21}$ defined likewise, one has $E_{12} E_{21} = E_{11} \\ne E_{22} = E_{21} E_{12}$.\nNon-example: the ring $\\Z$ is not noncommutative, since $ab = ba$ for all integers $a, b$.\nMany authors use \"noncommutative ring\" for a ring that is not assumed to be commutative.\nIn that use, commutative rings are allowed as special cases.\n",
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