opposite ring

If $R$ is a ring, then we may construct the opposite ring $R^{op}$ which has the same underlying abelian group structure, but with multiplication in the opposite order: the product of $r_{1}$ and $r_{2}$ in $R^{op}$ is $r_{2}r_{1}$ .

If $M$ is a left $R$ -module, then it can be made into a right $R^{op}$ -module, where a module element $m$ , when multiplied on the right by an element $r$ of $R^{op}$ , yields the $r m$ that we have with our left $R$ -module action on $M$ . Similarly, right $R$ -modules can be made into left $R^{op}$ -modules.

If $R$ is a commutative ring, then it is equal to its own opposite ring.

Similar constructions occur in the opposite group and opposite category.

单词	OppositeRing
释义	opposite ring If $R$ is a ring, then we may construct the opposite ring $R^{op}$ which has the same underlying abelian group structure, but with multiplication in the opposite order: the product of $r_{1}$ and $r_{2}$ in $R^{op}$ is $r_{2}r_{1}$ . If $M$ is a left $R$ -module, then it can be made into a right $R^{op}$ -module, where a module element $m$ , when multiplied on the right by an element $r$ of $R^{op}$ , yields the $r m$ that we have with our left $R$ -module action on $M$ . Similarly, right $R$ -modules can be made into left $R^{op}$ -modules. If $R$ is a commutative ring, then it is equal to its own opposite ring. Similar constructions occur in the opposite group and opposite category.
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