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 $$\mathfrak{c}$$,
 * $$\mathfrak{a} \cap (\mathfrak{b} + \mathfrak{c}) = (\mathfrak{a} \cap \mathfrak{b}) + (\mathfrak{a} \cap \mathfrak{c})$$
 * 1) 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.