multiplication ringMultiplicationRing2004-06-26 17:51:362007-02-02 04:00:26Definitionsum of idealsideal multiplication% this is the default PlanetMath preamble. as your knowledge
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\newtheorem*{thmplain}{Theorem}Let $R$ be a commutative ring with non-zero unity.\, If $\mathfrak{a}$ and $\mathfrak{b}$ are two \PMlinkname{\em fractional ideals}{FractionalIdealOfCommutativeRing} of $R$ with\, $\mathfrak{a} \subseteq \mathfrak{b}$\, and if $\mathfrak{b}$ is \PMlinkname{invertible}{FractionalIdealOfCommutativeRing}, then there is a \PMlinkescapetext{fractional ideal} $\mathfrak{c}$ of $R$ such that\, $\mathfrak{a} = \mathfrak{bc}$\, (one can choose\, $\mathfrak{c} = \mathfrak{b}^{-1}\mathfrak{a}$).
\textbf{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 {\em multiplication ring} if\, $\mathfrak{a} \subseteq \mathfrak{b}$\, always implies that there exists a \PMlinkescapetext{fractional ideal} $\mathfrak{c}$ of $R$ such that\, $\mathfrak{a} = \mathfrak{bc}$.
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\, Every Dedekind domain is a multiplication ring.\, If a multiplication ring has no zero divisors, it is a Dedekind domain.
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\bibitem{LM}{\sc M. Larsen \& P. McCarthy:} {\em Multiplicative theory of ideals}.\, Academic Press. New York (1971).
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