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 $R$-modulewhose 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 describesthe right module action of $R$ upon $M$,then $\varphi \alpha$ describesthe right module action of $R$ upon $M^{\alpha }$.

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