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.

Title	opposite ring
Canonical name	OppositeRing
Date of creation	2013-03-22 11:51:14
Last modified on	2013-03-22 11:51:14
Owner	antizeus (11)
Last modified by	antizeus (11)
Numerical id	7
Author	antizeus (11)
Entry type	Definition
Classification	msc 16B99
Classification	msc 17A01
Related topic	DualCategory
Related topic	NonCommutativeRingsOfOrderFour