multiplication ring

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

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

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