Arithmetical ring
# Arithmetical ring

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

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

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

