# 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 $\phi\colon R\to{\rm End}_{\mathbb{Z}}(M)$ 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:

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

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 $R\to\operatorname{End}_{\mathbb{Z}}(M)$, or switch to right notation for writing functions.

Title module Module1 2013-03-22 12:01:51 2013-03-22 12:01:51 yark (2760) yark (2760) 12 yark (2760) Definition msc 16D10 module action left module action right module action unital module Module