Arithmetical ring

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

The localization $R_{\mathfrak {m}}$ of R at ${\mathfrak {m}}$ is a uniserial ring for every maximal ideal ${\mathfrak {m}}$ of R.
For all ideals ${\mathfrak {a}},{\mathfrak {b}}$ , and ${\mathfrak {c}}$ ,
${\mathfrak {a}}\cap ({\mathfrak {b}}+{\mathfrak {c}})=({\mathfrak {a}}\cap {\mathfrak {b}})+({\mathfrak {a}}\cap {\mathfrak {c}})$
For all ideals ${\mathfrak {a}},{\mathfrak {b}}$ , and ${\mathfrak {c}}$ ,
${\mathfrak {a}}+({\mathfrak {b}}\cap {\mathfrak {c}})=({\mathfrak {a}}+{\mathfrak {b}})\cap ({\mathfrak {a}}+{\mathfrak {c}})$

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. ISSN 0092-7872. Zbl 1152.13015. doi:10.1080/00927870701351294.
Fuchs, Ladislas (1949). "Über die Ideale arithmetischer Ringe". Comment. Math. Helv. (in German). 23: 334–341. ISSN 0010-2571. Zbl 0040.30103. doi:10.1007/bf02565607.
Larsen, Max D.; McCarthy, Paul Joseph (1971). Multiplicative theory of ideals. Pure and Applied Mathematics. 43. Academic Press. pp. 150–151. ISBN 0080873561. Zbl 0237.13002.

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