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_\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.

References

External links

Arithmetical ring at PlanetMath.org.

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