multiplication ring

Let R be a commutative ring with non-zero unity.  If 𝔞 and 𝔟 are two fractional idealsMathworldPlanetmathPlanetmath ( of R with  ab  and if b is invertible (, then there is a c of R such that  a=bc  (one can choose  c=b-1a).

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  ab  always implies that there exists a c of R such that  a=bc.


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


  • 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