# unity

The unity of a ring $(R,+,\cdot )$ is the multiplicative identity^{} of the ring, if it has such. The unity is often denoted by $e$, $u$ or 1. Thus, the unity satisfies

$$e\cdot a=a\cdot e=a\mathit{\hspace{1em}}\forall a\in R.$$ |

If $R$ consists of the mere 0, then $0$ is its unity, since in every ring, $0\cdot a=a\cdot 0=0$. Conversely, if 0 is the unity in some ring $R$, then $R=\{0\}$ (because $a=0\cdot a=0\forall a\in R$).

Note. When considering a ring $R$ it is often mentioned that “…having $1\ne 0$” or that “…with non-zero unity”, sometimes only “…with unity” or “…with ”; all these exclude the case $R=\{0\}$.

###### Theorem.

An element $u$ of a ring $R$ is the unity iff $u$ is an idempotent^{} and regular element^{}.

Proof. Let $u$ be an idempotent and regular element. For any element $x$ of $R$ we have

$$ux={u}^{2}x=u(ux),$$ |

and because $u$ is no left zero divisor, it may be cancelled from the equation; thus we get $x=ux$. Similarly, $x=xu$. So $u$ is the unity of the ring. The other half of the is apparent.

Title | unity |

Canonical name | Unity |

Date of creation | 2013-03-22 14:47:17 |

Last modified on | 2013-03-22 14:47:17 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 15 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 20-00 |

Classification | msc 16-00 |

Classification | msc 13-00 |

Synonym | multiplicative identity |

Synonym | characterization of unity |

Related topic | ZeroDivisor |

Related topic | RootOfUnity |

Related topic | ZeroRing |

Related topic | NonZeroDivisorsOfFiniteRing |

Related topic | OppositePolynomial |

Defines | non-zero unity |

Defines | nonzero unity |