(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.
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.
|Date of creation||2013-03-22 12:01:51|
|Last modified on||2013-03-22 12:01:51|
|Last modified by||yark (2760)|
|Synonym||left module action|
|Synonym||right module action|