Arithmetical ring

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

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 $𝔠$,

   $𝔞\cap (𝔟+𝔠)=(𝔞\cap 𝔟)+(𝔞\cap 𝔠)$

3. For all ideals $𝔞,𝔟$, and $𝔠$,

   $𝔞+(𝔟\cap 𝔠)=(𝔞+𝔟)\cap (𝔞+𝔠)$

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.

References

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