(This is a definition of modules in terms of ring homomorphismsMathworldPlanetmath. You may prefer to read the other definition ( instead.)

Let R be a ring, and let M be an abelian groupMathworldPlanetmath.

We say that M is a left R-module if there exists a ring homomorphism ϕ:REnd(M) from R to the ring of abelian group endomorphismsPlanetmathPlanetmathPlanetmath on M (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 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 1R to the identityPlanetmathPlanetmathPlanetmathPlanetmath endomorphism on M, so that 1m=m for all mM. In this case we may say that the module is unital.

Typically the abelian group structureMathworldPlanetmath on M is expressed in additive terms, i.e. with operator +, identity elementMathworldPlanetmath 0M (or just 0), and inversesMathworldPlanetmathPlanetmathPlanetmath written in the form -m for mM.

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 REnd(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