Let $R$ be a ring, $M$ be an abelian group and $\cdot : R \times M \to M$. $M$ is a module over $R$ if:
(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 $x \in M$,
\[(\lambda + \mu)\cdot x = \lambda \cdot x + \mu \cdot x.\](Associativity) For all $\lambda$, $\mu \in R$ and $x \in M$,
\[\lambda \cdot (\mu \cdot x) = (\lambda \mu) \cdot x.\](Compatibility with $1_R$) For all $x \in M$,
\[1_R \cdot x = x.\]The operation $\cdot$ is then called a scalar multiplication.
Examples.
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Given any ring $R$, $\{0\}$ is a module over $R$ (the zero module).
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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).