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arithmetical ring
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(Theorem)
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Theorem 1 If $R$ is a commutative ring, then the following three conditions are equivalent:
- For all ideals $\mathfrak{a}$ , $\mathfrak{b}$ and $\mathfrak{c}$ of $R$ , one has $\mathfrak{a\cap(b+c) = (a\cap b)+(a\cap c)}$ .
- For all ideals $\mathfrak{a}$ , $\mathfrak{b}$ and $\mathfrak{c}$ of $R$ , one has $\mathfrak{a+(b\cap c) = (a+b)\cap(a+c)}$ .
- For each maximal ideal $\mathfrak{p}$ of $R$ the set of all ideals of $R_{\mathfrak{p}}$ , the localisation of $R$ at $R\!\setminus\!\mathfrak{p}$ , is totally ordered by set inclusion.
The ring $R$ satisfying the conditions of the theorem is called an arithmetical ring.
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"arithmetical ring" is owned by PrimeFan. [ owner history (2) ]
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Cross-references: theorem, ring, set inclusion, totally ordered, maximal ideal, ideals, equivalent, commutative ring
This is version 5 of arithmetical ring, born on 2005-07-18, modified 2006-10-29.
Object id is 7237, canonical name is ArithmeticalRing.
Accessed 3070 times total.
Classification:
| AMS MSC: | 13A99 (Commutative rings and algebras :: General commutative ring theory :: Miscellaneous) |
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Pending Errata and Addenda
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