# left and right unity of ring

If a ring $(R,+,\cdot )$ left identity^{} element $e$, i.e. if

$$e\cdot a=a\mathit{\hspace{1em}}\forall a,$$ |

then $e$ is called the left unity of $R$.

If a ring $R$ right identity element ${e}^{\prime}$, i.e. if

$$a\cdot {e}^{\prime}=a\mathit{\hspace{1em}}\forall a,$$ |

then ${e}^{\prime}$ is called the right unity of $R$.

A ring may have several left or right unities (see e.g. the Klein four-ring).

If a ring $R$ has both a left unity $e$ and a right unity ${e}^{\prime}$, then they must coincide, since

$${e}^{\prime}=e\cdot {e}^{\prime}=e.$$ |

This situation means that every right unity equals to $e$, likewise every left unity. Then we speak simply of a unity of the ring.

Title | left and right unity of ring |
---|---|

Canonical name | LeftAndRightUnityOfRing |

Date of creation | 2013-03-22 15:10:54 |

Last modified on | 2013-03-22 15:10:54 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 6 |

Author | rspuzio (6075) |

Entry type | Definition |

Classification | msc 20-00 |

Classification | msc 16-00 |

Related topic | InversesInRings |

Defines | left unity |

Defines | right unity |