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 be a ring,
and let be an abelian group.
We say that is a left -module
if there exists a ring homomorphism
from to the ring of abelian group endomorphisms on
(in which multiplication of endomorphisms is composition,
using left function notation).
We typically denote this function using a multiplication notation:
This ring homomorphism defines what is called a of upon .
If is a unital ring
(i.e. a ring with identity),
then we typically demand
that the ring homomorphism
map the unit
to the identity endomorphism on ,
so that for all .
In this case we may say
that the module is unital.
Typically the abelian group structure on
is expressed in additive terms,
i.e. with operator ,
identity element
(or just ),
and inverses
written in
the form for .
Right module actions are defined similarly, only with the elements of being written on the right sides of elements of . In this case we either need to use an anti-homomorphism , 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 |