arithmetical ring

If $R$ is a commutative ring, then the following three conditions are equivalent:

•
For all ideals $\\mathfrak{a}$ , $\\mathfrak{b}$ and $\\mathfrak{c}$ of $R$ , one has $\\mathfrak{a\\cap(b+c)=(a\\cap b)+(a\\cap c)}$ .
•
For all ideals $\\mathfrak{a}$ , $\\mathfrak{b}$ and $\\mathfrak{c}$ of $R$ , one has $\\mathfrak{a+(b\\cap c)=(a+b)\\cap(a+c)}$ .
•
For each maximal ideal $\\mathfrak{p}$ of $R$ the set of all ideals of $R_{\\mathfrak{p}}$ , the localisation (http://planetmath.org/Localization) of $R$ at $R\\!\\setminus\\!\\mathfrak{p}$ , is totally ordered by set inclusion.

The ring $R$ satisfying the conditions of the theorem is called an arithmetical ring.

单词	ArithmeticalRing
释义	arithmetical ring Theorem. If $R$ is a commutative ring, then the following three conditions are equivalent: • For all ideals $\\mathfrak{a}$ , $\\mathfrak{b}$ and $\\mathfrak{c}$ of $R$ , one has $\\mathfrak{a\\cap(b+c)=(a\\cap b)+(a\\cap c)}$ . • For all ideals $\\mathfrak{a}$ , $\\mathfrak{b}$ and $\\mathfrak{c}$ of $R$ , one has $\\mathfrak{a+(b\\cap c)=(a+b)\\cap(a+c)}$ . • For each maximal ideal $\\mathfrak{p}$ of $R$ the set of all ideals of $R_{\\mathfrak{p}}$ , the localisation (http://planetmath.org/Localization) of $R$ at $R\\!\\setminus\\!\\mathfrak{p}$ , is totally ordered by set inclusion. The ring $R$ satisfying the conditions of the theorem is called an arithmetical ring.
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