# uniqueness of additive identity in a ring

###### Lemma 1.

Let $R$ be a ring. There exists a unique element $\mathrm{0}$ in $R$ such that for all $a$ in $R$:

$$0+a=a+0=a.$$ |

###### Proof.

By the definition of ring, there exists at least one identity^{} in $R$, call it ${0}_{1}$. Suppose ${0}_{2}\in R$ is an element which also the of additive identity. Thus,

$${0}_{2}+{0}_{1}={0}_{2}$$ |

On the other hand, ${0}_{1}$ is an additive identity, therefore:

$${0}_{2}+{0}_{1}={0}_{1}+{0}_{2}={0}_{1}$$ |

Hence ${0}_{2}={0}_{1}$, i.e. there is a unique additive identity. ∎

Title | uniqueness of additive identity in a ring |
---|---|

Canonical name | UniquenessOfAdditiveIdentityInARing |

Date of creation | 2013-03-22 14:14:06 |

Last modified on | 2013-03-22 14:14:06 |

Owner | alozano (2414) |

Last modified by | alozano (2414) |

Numerical id | 6 |

Author | alozano (2414) |

Entry type | Theorem |

Classification | msc 13-00 |

Classification | msc 16-00 |

Classification | msc 20-00 |