title: Arithmetical ring
text: # Arithmetical ring
  
  In algebra, a commutative ring R is said to be arithmetical (or arithmetic) if any of the following equivalent conditions hold:
  
  * The localization $R_{\mathfrak {m}}$ of R at ${\mathfrak {m}}$ is a uniserial ring for every maximal ideal ${\mathfrak {m}}$ of R.
  * For all ideals ${\mathfrak {a}},{\mathfrak {b}}$, and ${\mathfrak {c}}$,
  
      ${\mathfrak {a}}\cap ({\mathfrak {b}}+{\mathfrak {c}})=({\mathfrak {a}}\cap {\mathfrak {b}})+({\mathfrak {a}}\cap {\mathfrak {c}})$
  * 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.
  
  