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).
随便看	AbelianNumberField abelian number field AbelianVariety abelian variety AbelPrize Abel Prize AbelsLemma AbelsLimitTheorem AbelsMultiplicationRuleForSeries AbelSummability Abel summability Abel’s lemma Abel’s limit theorem Abel’s multiplication rule for series ABIsConjugateToBA AB is conjugate to BA AbnormalSubgroup abnormal subgroup AbsoluteConvergenceImpliesConvergenceForAnInfiniteProduct absolute convergence implies convergence for an infinite product AbsoluteConvergenceImpliesUniformConvergence absolute convergence implies uniform convergence AbsoluteConvergenceOfDoubleSeries absolute convergence of double series AbsoluteConvergenceOfInfiniteProduct