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.
随便看	conjugate index ConjugateModule conjugate module ConjugatePoints conjugate points ConjugateStabilizerSubgroups conjugate stabilizer subgroups ConjugateTranspose conjugate transpose Conjugationmnemonic conjugation (mnemonic) Conjunction conjunction ConnectedComponent connected component ConnectedGraph connected graph ConnectedImKleinen connected im kleinen ConnectedLocallyCompactTopologicalGroupsAresigmacompact connected locally compact topological groups are σ-compact Connectedness ConnectednessIsPreservedUnderAContinuousMap connectedness is preserved under a continuous map Connectedness of suspensions