# module

(This is a definition of modules in terms of ring homomorphisms^{}. You may prefer to read the other definition (http://planetmath.org/Module) instead.)

Let $R$ be a ring,
and let $M$ be an abelian group^{}.

We say that $M$ is a left $R$-module
if there exists a ring homomorphism $\varphi :R\to {\mathrm{End}}_{\mathbb{Z}}(M)$
from $R$ to the ring of abelian group endomorphisms^{} on $M$
(in which multiplication of endomorphisms is composition,
using left function notation).
We typically denote this function using a multiplication notation:

$$[\varphi (r)](m)=r\cdot m=rm.$$ |

This ring homomorphism defines what is called a of $R$ upon $M$.

If $R$ is a unital ring
(i.e. a ring with identity),
then we typically demand
that the ring homomorphism
map the unit $1\in R$
to the identity^{} endomorphism on $M$,
so that $1\cdot m=m$ for all $m\in M$.
In this case we may say
that the module is *unital*.

Typically the abelian group structure^{} on $M$
is expressed in additive terms,
i.e. with operator $+$,
identity element^{} ${0}_{M}$ (or just $0$),
and inverses^{} written in
the form $-m$ for $m\in M$.

Right module actions are defined similarly, only with the elements of $R$ being written on the right sides of elements of $M$. In this case we either need to use an anti-homomorphism $R\to {\mathrm{End}}_{\mathbb{Z}}(M)$, or switch to right notation for writing functions.

Title | module |
---|---|

Canonical name | Module1 |

Date of creation | 2013-03-22 12:01:51 |

Last modified on | 2013-03-22 12:01:51 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 12 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 16D10 |

Synonym | module action |

Synonym | left module action |

Synonym | right module action |

Synonym | unital module |

Related topic | Module |