PlanetMath: multiplication ring

(more info)

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multiplication ring

(Definition)

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

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