arithmetical ring
Theorem.
If $R$ is a commutative ring, then the following three conditions are equivalent^{}:

β’
For all ideals $\mathrm{\pi \x9d\x94\x9e}$, $\mathrm{\pi \x9d\x94\x9f}$ and $\mathrm{\pi \x9d\x94}$ of $R$, one hasβ $\mathrm{\pi \x9d\x94\x9e}\beta \x88\copyright (\mathrm{\pi \x9d\x94\x9f}+\mathrm{\pi \x9d\x94})=(\mathrm{\pi \x9d\x94\x9e}\beta \x88\copyright \mathrm{\pi \x9d\x94\x9f})+(\mathrm{\pi \x9d\x94\x9e}\beta \x88\copyright \mathrm{\pi \x9d\x94})$.

β’
For all ideals $\mathrm{\pi \x9d\x94\x9e}$, $\mathrm{\pi \x9d\x94\x9f}$ and $\mathrm{\pi \x9d\x94}$ of $R$, one hasβ $\mathrm{\pi \x9d\x94\x9e}+(\mathrm{\pi \x9d\x94\x9f}\beta \x88\copyright \mathrm{\pi \x9d\x94})=(\mathrm{\pi \x9d\x94\x9e}+\mathrm{\pi \x9d\x94\x9f})\beta \x88\copyright (\mathrm{\pi \x9d\x94\x9e}+\mathrm{\pi \x9d\x94})$.

β’
For each maximal ideal^{} $\mathrm{\pi \x9d\x94\xad}$ of $R$ the set of all ideals of ${R}_{\mathrm{\pi \x9d\x94\xad}}$, the localisation (http://planetmath.org/Localization^{}) of $R$ atβ $R\beta \x88\x96\mathrm{\pi \x9d\x94\xad}$,β is totally ordered by set inclusion.
The ring $R$ satisfying the conditions of the theorem^{} is called an arithmetical ring.
Title  arithmetical ring 

Canonical name  ArithmeticalRing 
Date of creation  20130322 15:23:58 
Last modified on  20130322 15:23:58 
Owner  PrimeFan (13766) 
Last modified by  PrimeFan (13766) 
Numerical id  8 
Author  PrimeFan (13766) 
Entry type  Theorem 
Classification  msc 13A99 
Related topic  QuotientOfIdeals 