# additive inverse of an inverse element

In any ring $R$, the additive inverse of an element $a\in R$ must exist, is unique and is denoted by $-a$. Since $-a$ is also in the ring $R$ it also has an additive inverse in $R$, which is $-(-a)$. Put $-(-a)=c\in R$. Then by definition of the additive inverse, $-a+c=0$ and $-a+a=0$. Since additive inverses are unique, it must be that $c=a$.

Title | additive inverse of an inverse element |
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Canonical name | AdditiveInverseOfAnInverseElement |

Date of creation | 2013-03-22 15:45:16 |

Last modified on | 2013-03-22 15:45:16 |

Owner | Mathprof (13753) |

Last modified by | Mathprof (13753) |

Numerical id | 8 |

Author | Mathprof (13753) |

Entry type | Result |

Classification | msc 16B70 |

Related topic | InverseOfInverseInAGroup |