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).