Arithmetical ring


In algebra, a commutative ring R is said to be arithmetical if any of the following equivalent conditions holds:
  1. The localization of R at is a uniserial ring for every maximal ideal of R.
  2. For all ideals, and,
  3. :
  4. For all ideals, and,
  5. :
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.