conjugate module

If $M$ is a right module over a ring $R$, and $\alpha$ is an endomorphism of $R$, we define the conjugate module $M^{\alpha }$ to be the right $R$-module whose underlying set is $\{m^{\alpha }\mid m\in M\}$, with abelian group structure identical to that of $M$ (i.e. ${(m-n)}^{\alpha }=m^{\alpha }-n^{\alpha }$), and scalar multiplication given by $m^{\alpha }\cdot r={(m\cdot \alpha (r))}^{\alpha }$ for all $m$ in $M$ and $r$ in $R$.

In other words, if $\varphi :R\to {\mathrm{End}}_{\mathbb{Z}}(M)$ is the ring homomorphism that describes the right module action of $R$ upon $M$, then $\varphi \alpha$ describes the right module action of $R$ upon $M^{\alpha }$.

If $N$ is a left $R$-module, we define ${}^{\alpha }N$ similarly, with $r\cdot {}^{\alpha }n={}^{\alpha }(\alpha (r)\cdot n)$.

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