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# every ring is an integer algebra

Let $R$ be a ring. Then $R$ is also an algebra over the ring of integers if we define the action of $\mathbb{Z}$ on $R$ by the following rules:

$0\cdot x=0$ |

$(n+1)\cdot x=n\cdot x+x$ |

$(-n)\cdot x=-(n\cdot x)$ |

In other words, the action of a positive integer $n$ on $x$ is to add $x$ to itself $n$ times and the action of a negative integer $n$ on $x$ is to subtract $x$ to itself $n$ times.

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GeneralAssociativity

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## Mathematics Subject Classification

13B99*no label found*20C99

*no label found*16S99

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