Arithmetical ring

In algebra, a commutative 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". Commun. Algebra 35 (9): 2671–2684. doi:10.1080/00927870701351294. ISSN 0092-7872. 
• Fuchs, Ladislas (1949). "Über die Ideale arithmetischer Ringe" (in German). Comment. Math. Helv. 23: 334–341. doi:10.1007/bf02565607. ISSN 0010-2571. 
• Larsen, Max D.; McCarthy, Paul Joseph (1971). Multiplicative theory of ideals. Pure and Applied Mathematics. 43. Academic Press. pp. 150–151. ISBN 0080873561. 

External links

"Arithmetical ring". http://planetmath.org/?op=getobj&from=objects&id={{{id}}}.