opposite ring

If R is a ring, then we may construct the opposite ring Rop which has the same underlying abelian groupMathworldPlanetmath structureMathworldPlanetmath, but with multiplication in the opposite order: the product of r1 and r2 in Rop is r2r1.

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