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arithmetical ring
Theorem.
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.
Related:
QuotientOfIdeals
Major Section:
Reference
Type of Math Object:
Theorem
Parent:
Mathematics Subject Classification
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