multiplication ring
Let be a commutative ring with non-zero unity. If and are two fractional ideals (http://planetmath.org/FractionalIdealOfCommutativeRing) of with and if is invertible (http://planetmath.org/FractionalIdealOfCommutativeRing), then there is a of such that (one can choose ).
Definition. Let be a commutative ring with non-zero unity and let and be ideals of . The ring is a multiplication ring if always implies that there exists a of such that .
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 |
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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 |