# quotient module

Let $M$ be a module over a ring $R$, and let $S$ be a submodule of $M$.
The *quotient module* $M/S$ is the quotient group^{} $M/S$ with
scalar multiplication defined by $\lambda (x+S)=\lambda x+S$ for all
$\lambda \in R$ and all $x\in M$.

This is a well defined operation. Indeed, if $x+S={x}^{\prime}+S$ then for some $s\in S$ we have ${x}^{\prime}=x+s$ and therefore

$\lambda {x}^{\prime}$ | $=\lambda (x+s)$ | ||

$=\lambda x+\lambda s$ |

so that $\lambda {x}^{\prime}+S=\lambda x+\lambda s+S=\lambda x+S$, since $\lambda s\in S$.

In the special case that $R$ is a field this construction defines
the *quotient vector space* of a vector space^{} by a vector subspace.

Title | quotient module |
---|---|

Canonical name | QuotientModule |

Date of creation | 2013-03-22 14:01:18 |

Last modified on | 2013-03-22 14:01:18 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 9 |

Author | rspuzio (6075) |

Entry type | Definition |

Classification | msc 16D10 |

Defines | quotient vector space |