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.