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

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

   ${\\displaystyle {\\mathfrak {a}}\\cap ({\\mathfrak {b}}+{\\mathfrak {c}})=({\\mathfrak {a}}\\cap {\\mathfrak {b}})+({\\mathfrak {a}}\\cap {\\mathfrak {c}})}$

3. For all ideals ${\\displaystyle {\\mathfrak {a}},{\\mathfrak {b}}}$, and ${\\displaystyle {\\mathfrak {c}}}$,

   ${\\displaystyle {\\mathfrak {a}}+({\\mathfrak {b}}\\cap {\\mathfrak {c}})=({\\mathfrak {a}}+{\\mathfrak {b}})\\cap ({\\mathfrak {a}}+{\\mathfrak {c}})}$