Arithmetical ring

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

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

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

3. 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. 

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

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

External links

• "Arithmetical ring". https://planetmath.org/ArithmeticalRing.