conjugate module

If M is a right module over a ring R, and α is an endomorphismPlanetmathPlanetmathPlanetmath of R, we define the conjugate module Mα to be the right R-module whose underlying set is {mαmM}, with abelian groupMathworldPlanetmath structureMathworldPlanetmath identical to that of M (i.e. (m-n)α=mα-nα), and scalar multiplication given by mαr=(mα(r))α for all m in M and r in R.

In other words, if ϕ:REnd(M) is the ring homomorphismMathworldPlanetmath that describes the right module action of R upon M, then ϕα describes the right module action of R upon Mα.

If N is a left R-module, we define Nα similarly, with rnα=(α(r)n)α.

Title conjugate module
Canonical name ConjugateModule
Date of creation 2013-03-22 11:49:47
Last modified on 2013-03-22 11:49:47
Owner antizeus (11)
Last modified by antizeus (11)
Numerical id 9
Author antizeus (11)
Entry type Definition
Classification msc 16D10
Classification msc 41A45