Module

Let $R$ be a ring (often assumed a unital ring ). A left $R$-module is an abelian group $(M,+)$ together with a scalar multiplication map $R\times M\to M$, $(r,m)\mapsto rm$, such that for all $r,s\in R$ and $m,n\in M$:

1. $r(m+n)=rm+rn$,
2. $(r+s)m=rm+sm$,
3. $(rs)m=r(sm)$,
4. if $R$ is unital, then $1_R m=m$.

A right $R$-module is defined similarly with a map $M\times R\to M$, $(m,r)\mapsto mr$, satisfying the analogous axioms.

The axioms are collected in module axioms . When $R$ is a field , left $R$-modules are the same objects as vector spaces . Ideals of a ring give basic examples of modules, linking module theory to ideal theory.

Examples:

• For any ring $R$, the additive group of $R$ is a left $R$-module via multiplication: $r\cdot x=rx$.
• For $R=\mathbb Z$, a left $\mathbb Z$-module is exactly an abelian group (with $n\cdot m$ defined as repeated addition).
• If $I\lhd R$ is an ideal, then $I$ is an $R$-module under the restricted multiplication action.