# 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.

Title | every ring is an integer algebra |
---|---|

Canonical name | EveryRingIsAnIntegerAlgebra |

Date of creation | 2013-03-22 14:47:47 |

Last modified on | 2013-03-22 14:47:47 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 5 |

Author | rspuzio (6075) |

Entry type | Example |

Classification | msc 13B99 |

Classification | msc 20C99 |

Classification | msc 16S99 |

Related topic | GeneralAssociativity |