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 $rm$ 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.