# 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.)

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.

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