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Arithmetical ring

From Wikipedia, the free encyclopedia

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

  1. The localization of R at is a uniserial ring for every maximal ideal of R.
  2. For all ideals , and ,
  3. For all ideals , and ,

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

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  • 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.
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"Arithmetical ring". PlanetMath.