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).

单词	MultiplicationRing
释义	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).
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