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[parent] quotient of ideals (Definition)

Let $ R$ be a commutative ring having regular elements and let $ T$ be its total ring of fractions. If $ \mathfrak{a}$ and $ \mathfrak{b}$ are fractional ideals of $ R$, then one can define two different quotients or residuals of $ \mathfrak{a}$ by $ \mathfrak{b}$:

  • $ \mathfrak{a\!:\!b}\, := \{r\in R\vert\quad r\mathfrak{b} \subseteq \mathfrak{a}\}$
  • $ [\mathfrak{a\!:\!b}] := \{t\in T\vert\quad t\mathfrak{b} \subseteq \mathfrak{a}\}$

They both are fractional ideals of $ R$, and the former in fact an integral ideal of $ R$. It is clear that

$\displaystyle \mathfrak{a\!:\!b} = [\mathfrak{a\!:\!b}]\cap\!R.$
In the special case that $ R$ has non-zero unity and $ \mathfrak{b}$ has the inverse ideal $ \mathfrak{b}^{-1}$, we have
$\displaystyle [\mathfrak{a\!:\!b}] = \mathfrak{a}\mathfrak{b}^{-1},$
in particular
$\displaystyle [R\!:\!\mathfrak{b}] = \mathfrak{b}^{-1}.$

Some rules concerning the former type of quotient (the corresponding rules are valid also for the latter type):

  1. $ \mathfrak{a}\subseteq\mathfrak{b}\,\,\,\,\Rightarrow\,\,\, \mathfrak{a}:\mathf... ...rak{c}\,\, \land\,\,\mathfrak{c}:\mathfrak{a}\supseteq\mathfrak{c}:\mathfrak{b}$
  2. $ \mathfrak{a}:(\mathfrak{b}\mathfrak{c}) = (\mathfrak{a}:\mathfrak{b}):\mathfrak{c}$
  3. $ \mathfrak{a}:(\mathfrak{b}+\mathfrak{c})= (\mathfrak{a}:\mathfrak{b})\cap(\mathfrak{a}:\mathfrak{c})$
  4. $ (\mathfrak{a}\cap\mathfrak{b}):\mathfrak{c} = (\mathfrak{a}:\mathfrak{c})\cap(\mathfrak{b}:\mathfrak{c})$
Remark. In a Prüfer ring $ R$ the addition and intersection of ideals are dual operations of each other in the sense that there we have the duals

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

$ \quad\quad (\mathfrak{a}+\mathfrak{b}):\mathfrak{c} = (\mathfrak{a}:\mathfrak{c})+(\mathfrak{b}:\mathfrak{c})$

of the two last rules if the divisor ideals are finitely generated.



"quotient of ideals" is owned by pahio.
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See Also: sum of ideals, product of ideals, submodule, arithmetical ring

Other names:  residual, quotient ideal
Keywords:  fractional ideal

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Cross-references: finitely generated, operations, ideals, intersection, Prüfer ring, quotient, inverse ideal, non-zero unity, clear, integral ideal, fractional ideals, total ring of fractions, regular elements, commutative ring
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This is version 16 of quotient of ideals, born on 2004-11-11, modified 2005-07-19.
Object id is 6468, canonical name is QuotientOfIdeals.
Accessed 3838 times total.

Classification:
AMS MSC13B30 (Commutative rings and algebras :: Ring extensions and related topics :: Quotients and localization)

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