Английская Википедия:Arithmetical ring

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

1. The localization <math>R_\mathfrak{m}</math> of R at <math>\mathfrak{m}</math> is a uniserial ring for every maximal ideal <math>\mathfrak{m}</math> of R.
2. For all ideals <math>\mathfrak{a}, \mathfrak{b}</math>, and <math>\mathfrak{c}</math>,

   <math>\mathfrak{a} \cap (\mathfrak{b} + \mathfrak{c}) = (\mathfrak{a} \cap \mathfrak{b}) + (\mathfrak{a} \cap \mathfrak{c})</math>

3. For all ideals <math>\mathfrak{a}, \mathfrak{b}</math>, and <math>\mathfrak{c}</math>,

   <math>\mathfrak{a} + (\mathfrak{b} \cap \mathfrak{c}) = (\mathfrak{a} + \mathfrak{b}) \cap (\mathfrak{a} + \mathfrak{c})</math>

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

• Шаблон:Cite journal
• Шаблон:Cite journal
• Шаблон:Cite book