Arithmetical ring

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

1. The localization R m of R at m is a uniserial ring for every maximal ideal m of R.
2. For all ideals a , b , and c , a ∩ ( b + c ) = ( a ∩ b ) + ( a ∩ c )
3. 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.