# inverses in rings

Let $R$ be a ring with unity $1$ and $r\in R$. Then $r$ is *left invertible* if there exists $q\in R$ with $qr=1$; $q$ is a *left inverse ^{}* of $r$. Similarly, $r$ is

*right invertible*if there exists $s\in R$ with $rs=1$; $s$ is a

*right inverse*of $r$.

Note that, if $r$ is left invertible, it may not have a unique left inverse, and similarly for right invertible elements. On the other hand, if $r$ is left invertible and right invertible, then it has exactly one left inverse and one right inverse. Moreover, these two are equal, and $r$ is a unit.

Title | inverses in rings |

Canonical name | InversesInRings |

Date of creation | 2013-03-22 17:08:55 |

Last modified on | 2013-03-22 17:08:55 |

Owner | Wkbj79 (1863) |

Last modified by | Wkbj79 (1863) |

Numerical id | 4 |

Author | Wkbj79 (1863) |

Entry type | Topic |

Classification | msc 16-00 |

Related topic | Klein4Ring |

Related topic | LeftAndRightUnityOfRing |

Defines | left invertible |

Defines | right invertible |

Defines | left inverse |

Defines | right inverse |