# 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 $rm$ 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.

Title opposite ring OppositeRing 2013-03-22 11:51:14 2013-03-22 11:51:14 antizeus (11) antizeus (11) 7 antizeus (11) Definition msc 16B99 msc 17A01 DualCategory NonCommutativeRingsOfOrderFour