# 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 |