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  "body": "\n## Summary\n\nThe opposite ring of a [[ring|0B1261B5ED]] $R$ has the same elements and the same addition as $R$, with the factors of every product swapped.\nA ring equals its opposite exactly when it is commutative.\nThe opposite ring turns right modules into left modules and back.\nSo results proved for one side carry over to the other.\nIt also appears in the endomorphism ring of a ring viewed as a module over itself.\n\n## Explanations\n\n### L1 human\n\nTake a ring and read every product backwards: the new product of $a$ and $b$ is the old product $ba$.\nThe result is again a ring, called the opposite ring.\nFor commutative rings nothing changes.\nFor matrices over a commutative ring such as $\\R$, the transpose turns $AB$ into $B^T A^T$, so the transpose map identifies the matrix ring with its opposite.\n",
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