Module

Pub Apr 15, 2026 Up Jun 20, 2026

Definition 1. Let $R$ be a ring, $M$ be an abelian group and $\cdot : R \times M \to M$. $M$ is a module over $R$ if:

  1. (Distributivity) For all $\lambda \in R$ and $x$, $y \in M$,

    \[\lambda \cdot (x + y) = \lambda \cdot x + \lambda \cdot y;\]

    for all $\lambda$, $\mu \in R$ and $x \in M$,

    \[(\lambda + \mu)\cdot x = \lambda \cdot x + \mu \cdot x.\]

  2. (Associativity) For all $\lambda$, $\mu \in R$ and $x \in M$,

    \[\lambda \cdot (\mu \cdot x) = (\lambda \mu) \cdot x.\]

  3. (Compatibility with $1_R$) For all $x \in M$,

    \[1_R \cdot x = x.\]

The operation $\cdot$ is then called a scalar multiplication.

Examples.

  1. Given any ring $R$, $\{0\}$ is a module over $R$ (the zero module).

  2. Given any ring $R$ and $n \in \N^+$, $R^n$ is a module with scalar multiplication defined by

    \[\lambda \cdot(x_1, \cdots, x_n) = (\lambda x_1, \cdots, \lambda x_n)\]

    for all $\lambda \in R$ and $(x_1, \cdots, x_n) \in R^n$ (the standard module).