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'multiplication ring'
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| Title of object: |
multiplication ring |
| Canonical Name: |
MultiplicationRing |
| Type: |
Definition |
| Created on: |
2004-06-26 17:51:36 |
| Modified on: |
2004-10-31 09:37:14 |
| Classification: |
msc:13A15 |
| Keywords: |
ideal multiplication |
Preamble:
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Content:
Let $R$ be a commutative ring with non-zero unity. \,If $\mathfrak{a}$ and $\mathfrak{b}$ are two \PMlinkescapetext{{\em fractional ideals}}\footnote{The notion of \PMlinkescapetext{``fractional ideal''} used here is the one defined in the entry ``Pr\"ufer ring''.} of $R$ with \,$\mathfrak{a} \subseteq \mathfrak{b}$ and if $\mathfrak{b}$ is invertible, 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 \,$\mathfrak{a} = \mathfrak{bc}$.
\textbf{Theorem.} \,Every Dedekind domain is a multiplication ring. \,If a multiplication ring has no zero divisors, it is a Dedekind domain.
\begin{thebibliography}{9}
\bibitem{Larsen & McCarthy} M. Larsen and P. McCarthy: {\em Multiplicative theory of ideals}. \,Academic Press. New York (1971).
\end{thebibliography} |
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