Arithmetical ring

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

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". Communications in Algebra. 35 (9): 2671–2684. doi:10.1080/00927870701351294. Zbl 1152.13015.

Fuchs, Ladislaus (1949). "Über die Ideale arithmetischer Ringe". Commentarii Mathematici Helvetici (in German). 23: 334–341. doi:10.1007/bf02565607. Zbl 0040.30103.

Larsen, Max D.; McCarthy, Paul J. (1971). Multiplicative Theory of Ideals. Pure and Applied Mathematics. Vol. 43. New York: Academic Press. pp. 150–151. ISBN 978-0-08087356-5. Zbl 0237.13002.