# 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 $\phi:R\to{\rm End}_{\mathbb{Z}}(M)$ is the ring homomorphism that describes the right module action of $R$ upon $M$, then $\phi\alpha$ 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 ConjugateModule 2013-03-22 11:49:47 2013-03-22 11:49:47 antizeus (11) antizeus (11) 9 antizeus (11) Definition msc 16D10 msc 41A45