# 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 (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.

Title arithmetical ring ArithmeticalRing 2013-03-22 15:23:58 2013-03-22 15:23:58 PrimeFan (13766) PrimeFan (13766) 8 PrimeFan (13766) Theorem msc 13A99 QuotientOfIdeals