A module is a type of object which appears frequently in abstract algebra. It is a generalization of the concept of a vector space.

Specifically, given a ring $R$ a (left) $R$-module is an abelian group $(M,+)$ together with an operation $R\times M\to M$ (called scalar multiplication) written as $r\cdot x$ or $rx$, which satisfies the following properties:

For all $a,b\in R$, $x,y\in M$

(1) $(a+b)x = ax+bx$

(2) $a(x+y) = ax+ay$

(3) $a(bx) = (ab)x$

(4) $1x = x$

We typically write $M$ to mean the module as well as the underlying abelian group.

If $R$ is a field then $M$ is a vector space over $R$.

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