# unity of subring

###### Theorem.

Let $S$ be a proper subring of the ring $R$. If $S$ has a non-zero unity $u$ which is not unity of $R$, then $u$ is a zero divisor^{} of $R$.

Proof. Because $u$ is not unity of $R$, there exists an element $r$ of $R$ such that $ru\ne r$. Then we have $(ru)u=r(uu)=ru$, which implies that $0=(ru)u-ru=(ru-r)\cdot u$. Since neither $ru-r$ nor $u$ is 0, the element $u$ is a zero divisor in $R$.

Title | unity of subring |
---|---|

Canonical name | UnityOfSubring |

Date of creation | 2013-03-22 14:49:40 |

Last modified on | 2013-03-22 14:49:40 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 7 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 20-00 |

Classification | msc 16-00 |

Classification | msc 13-00 |

Related topic | UnitiesOfRingAndSubring |

Related topic | CornerOfARing |