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[parent] multiplication ring (Definition)

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

Theorem 1   Every Dedekind domain is a multiplication ring. If a multiplication ring has no zero divisors, it is a Dedekind domain.

Bibliography

1
M. LARSEN & P. MCCARTHY: Multiplicative theory of ideals. Academic Press. New York (1971).




"multiplication ring" is owned by PrimeFan. [ owner history (2) ]
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See Also: Prüfer ring, Dedekind domain, divisibility in rings

Keywords:  ideal multiplication

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Cross-references: zero divisors, Dedekind domain, implies, ring, ideals, non-zero unity, commutative ring
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This is version 14 of multiplication ring, born on 2004-06-26, modified 2007-02-02.
Object id is 5967, canonical name is MultiplicationRing.
Accessed 2564 times total.

Classification:
AMS MSC13A15 (Commutative rings and algebras :: General commutative ring theory :: Ideals; multiplicative ideal theory)

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