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

- The localization
R
m
of
*R* at
m
is a uniserial ring for every maximal ideal
m
of *R*.
- For all ideals
a
,
b
, and
c
,
a
∩
(
b
+
c
)
=
(
a
∩
b
)
+
(
a
∩
c
)
- For all ideals
a
,
b
, and
c
,
a
+
(
b
∩
c
)
=
(
a
+
b
)
∩
(
a
+
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.