# multiplication ring

Let $R$ be a commutative ring with non-zero unity.  If $\mathfrak{a}$ and $\mathfrak{b}$ are two 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 MultiplicationRing 2013-03-22 14:27:02 2013-03-22 14:27:02 PrimeFan (13766) PrimeFan (13766) 17 PrimeFan (13766) Definition msc 13A15 PruferRing DedekindDomain DivisibilityInRings