module

Let $R$ be a ring with identity. A left module $M$ over $R$ is a set with two binary operations, $+: M\times M \longrightarrow M$ and $\cdot: R \times M \longrightarrow M$ , such that

1. $(\u+\v)+\w = \u+(\v+\w)$ for all $\u,\v,\w \in M$
2. $\u+\v=\v+\u$ for all $\u,\v\in M$
3. There exists an element $\0 \in M$ such that $\u+\0=\u$ for all $\u \in M$
4. For any $\u \in M$ , there exists an element $\v \in M$ such that $\u+\v=\0$
5. $a \cdot (b \cdot \u) = (a \cdot b) \cdot \u$ for all $a,b \in R$ and $\u \in M$
6. $a \cdot (\u+\v) = (a \cdot \u) + (a \cdot \v)$ for all $a \in R$ and $\u,\v \in M$
7. $(a + b) \cdot \u = (a \cdot \u) + (b \cdot \u)$ for all $a,b \in R$ and $\u \in M$

A left module $M$ over $R$ is called unitary or unital if $1_R \cdot \u = \u$ for all $\u \in M$ .

A (unitary or unital) right module is defined analogously, except that the function $\cdot$ goes from $M \times R$ to $M$ and the scalar multiplication operations act on the right. If $R$ is commutative, there is an equivalence of categories between the category of left $R$ -modules and the category of right $R$ -modules.