Module

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:

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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