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.

• 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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