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 [math]\displaystyle{ R_\mathfrak{m} }[/math] of R at [math]\displaystyle{ \mathfrak{m} }[/math] is a uniserial ring for every maximal ideal [math]\displaystyle{ \mathfrak{m} }[/math] of R.
2. For all ideals [math]\displaystyle{ \mathfrak{a}, \mathfrak{b} }[/math], and [math]\displaystyle{ \mathfrak{c} }[/math],

   [math]\displaystyle{ \mathfrak{a} \cap (\mathfrak{b} + \mathfrak{c}) = (\mathfrak{a} \cap \mathfrak{b}) + (\mathfrak{a} \cap \mathfrak{c}) }[/math]

3. For all ideals [math]\displaystyle{ \mathfrak{a}, \mathfrak{b} }[/math], and [math]\displaystyle{ \mathfrak{c} }[/math],

   [math]\displaystyle{ \mathfrak{a} + (\mathfrak{b} \cap \mathfrak{c}) = (\mathfrak{a} + \mathfrak{b}) \cap (\mathfrak{a} + \mathfrak{c}) }[/math]

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}}}.