PlanetMath: quotient of ideals

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quotient of ideals

(Definition)

Let be a commutative ring having regular elements and let be its total ring of fractions. If $\mathfrak{a}$ and $\mathfrak{b}$ are fractional ideals of , 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 , and the former in fact an integral ideal of . It is clear that

$\displaystyle \mathfrak{a\!:\!b} = [\mathfrak{a\!:\!b}]\cap\!R.$

In the special case that

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):

$\mathfrak{a}\subseteq\mathfrak{b}\,\,\,\,\Rightarrow\,\,\, \mathfrak{a}:\mathf... ...rak{c}\,\, \land\,\,\mathfrak{c}:\mathfrak{a}\supseteq\mathfrak{c}:\mathfrak{b}$
$\mathfrak{a}:(\mathfrak{b}\mathfrak{c}) = (\mathfrak{a}:\mathfrak{b}):\mathfrak{c}$
$\mathfrak{a}:(\mathfrak{b}+\mathfrak{c})= (\mathfrak{a}:\mathfrak{b})\cap(\mathfrak{a}:\mathfrak{c})$
$(\mathfrak{a}\cap\mathfrak{b}):\mathfrak{c} = (\mathfrak{a}:\mathfrak{c})\cap(\mathfrak{b}:\mathfrak{c})$

Remark. In a Prüfer ring

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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Other names:

residual, quotient ideal

Keywords:

fractional ideal

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13B30 (Commutative rings and algebras :: Ring extensions and related topics :: Quotients and localization)

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