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