multiplication ring


Let R be a commutative ring with non-zero unity.  If π”ž and π”Ÿ are two fractional idealsMathworldPlanetmathPlanetmath (http://planetmath.org/FractionalIdealOfCommutativeRing) of R with  aβŠ†b  and if b is invertible (http://planetmath.org/FractionalIdealOfCommutativeRing), then there is a c of R such that  a=b⁒c  (one can choose  c=b-1⁒a).

Definition.  Let R be a commutative ring with non-zero unity and let a and b be ideals of R.  The ring R is a multiplication ring if  aβŠ†b  always implies that there exists a c of R such that  a=b⁒c.

Theorem.

Every Dedekind domainMathworldPlanetmath is a multiplication ring.  If a multiplication ring has no zero divisorsMathworldPlanetmath, 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