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In algebra, a commutative ring R is said to be arithmetical (or arithmetic) if any of the following equivalent conditions hold:

1. The localization ${\displaystyle R_{\mathfrak {m}}}$ of R at ${\displaystyle {\mathfrak {m}}}$ is a uniserial ring for every maximal ideal ${\displaystyle {\mathfrak {m}}}$ of R.
2. For all ideals ${\displaystyle {\mathfrak {a}},{\mathfrak {b}}}$, and ${\displaystyle {\mathfrak {c}}}$,

   ${\displaystyle {\mathfrak {a}}\cap ({\mathfrak {b}}+{\mathfrak {c}})=({\mathfrak {a}}\cap {\mathfrak {b}})+({\mathfrak {a}}\cap {\mathfrak {c}})}$

3. For all ideals ${\displaystyle {\mathfrak {a}},{\mathfrak {b}}}$, and ${\displaystyle {\mathfrak {c}}}$,

   ${\displaystyle {\mathfrak {a}}+({\mathfrak {b}}\cap {\mathfrak {c}})=({\mathfrak {a}}+{\mathfrak {b}})\cap ({\mathfrak {a}}+{\mathfrak {c}})}$

The last two conditions both say that the lattice of all ideals of R is distributive.

An arithmetical domain is the same thing as a Prüfer domain.

References

• Boynton, Jason (2007). "Pullbacks of arithmetical rings". Commun. Algebra. 35 (9): 2671–2684. doi:10.1080/00927870701351294. ISSN 0092-7872. S2CID 120927387. Zbl 1152.13015.
• Fuchs, Ladislas (1949). "Über die Ideale arithmetischer Ringe". Comment. Math. Helv. (in German). 23: 334–341. doi:10.1007/bf02565607. ISSN 0010-2571. S2CID 121260386. Zbl 0040.30103.
• Larsen, Max D.; McCarthy, Paul Joseph (1971). Multiplicative theory of ideals. Pure and Applied Mathematics. Vol. 43. Academic Press. pp. 150–151. ISBN 0080873561. Zbl 0237.13002.

External links

"Arithmetical ring". PlanetMath.

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