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  "body": "\n## Definition\n\nLet $(R, +, \\cdot)$ be a [[ring|0894J980RK]].\nThe *opposite ring* $R^{\\mathrm{op}}$ is the set $R$ with the same addition and the product $*$ given by\n$$a * b = b \\cdot a$$\nfor all $a, b \\in R$.\n\n## Remarks\n\n$R^{\\mathrm{op}}$ is a ring with the same zero and the same unit element as $R$.\nAssociativity holds because $(a * b) * c = c \\cdot (b \\cdot a) = (c \\cdot b) \\cdot a = a * (b * c)$.\nExample: for a commutative ring $K$, the transpose map $A \\mapsto A^T$ is a ring isomorphism $M_n(K) \\to M_n(K)^{\\mathrm{op}}$, since $(AB)^T = B^T A^T = A^T * B^T$.\nNon-example: for a nonzero commutative ring $K$, the identity map of $M_2(K)$ is not a ring isomorphism $M_2(K) \\to M_2(K)^{\\mathrm{op}}$, because it would need $AB = BA$ for all $A, B$.\nA right $R$-module is the same thing as a left $R^{\\mathrm{op}}$-module, with $r \\cdot m = m r$.\n",
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