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'ideal inverting in Pr\"ufer ring'
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| Title of object: |
ideal inverting in Pr\"ufer ring |
| Canonical Name: |
IdealInvertingInPruferRing |
| Type: |
Definition |
| Created on: |
2004-08-15 15:24:55 |
| Modified on: |
2004-08-15 15:32:59 |
| Classification: |
msc:13C13 |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
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\usepackage{amssymb}
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%\usepackage{psfrag}
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Content:
\textbf{Theorem.} \,Let $\mathfrak{a}_1$, ..., $\mathfrak{a}_n$ be invertible fractional ideals of a Pr\"ufer ring. \,Then also their sum and intersection are invertible, and the inverse ideals of these are obtained by the formulae resembling de Morgan's laws:
(\mathfrak{a}_1+...+\mathfrak{a}_n)^{-1} = \mathfrak{a}_1^{-1}\cap...\cap\mathfrak{a}_n^{-1}
(\mathfrak{a}_1\cap...\cap\mathfrak{a}_n)^{-1} = \mathfrak{a}_1^{-1}+...+\mathfrak{a}_n^{-1}
$$ |
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