# multiplication ring

Let $R$ be a commutative ring with non-zero unity. If $\U0001d51e$ and $\U0001d51f$ are two fractional ideals^{} (http://planetmath.org/FractionalIdealOfCommutativeRing) of $R$ with $\mathrm{a}\mathrm{\subseteq}\mathrm{b}$ and if $\mathrm{b}$ is invertible (http://planetmath.org/FractionalIdealOfCommutativeRing), then there is a $\mathrm{c}$ of $R$ such that $\mathrm{a}\mathrm{=}\mathrm{b}\mathit{}\mathrm{c}$ (one can choose $\mathrm{c}\mathrm{=}{\mathrm{b}}^{\mathrm{-}\mathrm{1}}\mathit{}\mathrm{a}$).

Definition. Let $R$ be a commutative ring with non-zero unity and let $\mathrm{a}$ and $\mathrm{b}$ be ideals of $R$. The ring $R$ is a multiplication ring if $\mathrm{a}\mathrm{\subseteq}\mathrm{b}$ always implies that there exists a $\mathrm{c}$ of $R$ such that $\mathrm{a}\mathrm{=}\mathrm{b}\mathit{}\mathrm{c}$.

###### Theorem.

Every Dedekind domain^{} is a multiplication ring. If a multiplication ring has no zero divisors^{}, it is a Dedekind domain.

## References

- 1 M. Larsen & P. McCarthy: Multiplicative theory of ideals. Academic Press. New York (1971).

Title | multiplication ring |
---|---|

Canonical name | MultiplicationRing |

Date of creation | 2013-03-22 14:27:02 |

Last modified on | 2013-03-22 14:27:02 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 17 |

Author | PrimeFan (13766) |

Entry type | Definition |

Classification | msc 13A15 |

Related topic | PruferRing |

Related topic | DedekindDomain |

Related topic | DivisibilityInRings |