# topological ring

A ring $R$ which is a topological space^{} is called a *topological ring* if the addition, multiplication, and the additive inverse functions are continuous functions^{} from $R\times R$ to $R$.

A *topological division ring* is a topological ring such that the multiplicative inverse^{} function is continuous away from $0$. A *topological field* is a topological division ring that is a field.

Remark. It is easy to see that if $R$ contains the multiplicative identity $1$, then $R$ is a topological ring iff addition and multiplication are continuous. This is true because the additive inverse of an element can be written as the product of the element and $-1$. However, if $R$ does not contain $1$, it is necessary to impose the continuity condition on the additive inverse operation.

Title | topological ring |

Canonical name | TopologicalRing |

Date of creation | 2013-03-22 12:45:59 |

Last modified on | 2013-03-22 12:45:59 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 6 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 12J99 |

Classification | msc 13J99 |

Classification | msc 54H13 |

Related topic | TopologicalGroup |

Related topic | TopologicalVectorSpace |

Defines | topological field |

Defines | topological division ring |