arithmetical ring

If $R$ is a commutative ring, then the following three conditions are equivalent:

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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 (http://planetmath.org/Localization) 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.