# quotient of ideals

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 or residuals of $\mathfrak{a}$ by $\mathfrak{b}$:

• $\mathfrak{a\!:\!b}\,\;:=\;\{r\in R|\quad r\mathfrak{b}\subseteq\mathfrak{a}\}$

• $[\mathfrak{a\!:\!b}]\;:=\;\{t\in T|\quad t\mathfrak{b}\subseteq\mathfrak{a}\}$

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

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

In the special case that $R$ has non-zero unity and $\mathfrak{b}$ has the inverse ideal $\mathfrak{b}^{-1}$, we have

 $[\mathfrak{a\!:\!b}]\;=\;\mathfrak{a}\mathfrak{b}^{-1},$

in particular

 $[R\!:\!\mathfrak{b}]\;=\;\mathfrak{b}^{-1}.$

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

1. 1.

$\mathfrak{a}\subseteq\mathfrak{b}\,\,\,\,\Rightarrow\,\,\,\mathfrak{a}:% \mathfrak{c}\subseteq\mathfrak{b}:\mathfrak{c}\,\,\land\,\,\mathfrak{c}:% \mathfrak{a}\supseteq\mathfrak{c}:\mathfrak{b}$

2. 2.

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

3. 3.

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

4. 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 (http://planetmath.org/SumOfIdeals) 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 are finitely generated.

 Title quotient of ideals Canonical name QuotientOfIdeals Date of creation 2013-03-22 14:48:36 Last modified on 2013-03-22 14:48:36 Owner pahio (2872) Last modified by pahio (2872) Numerical id 20 Author pahio (2872) Entry type Definition Classification msc 13B30 Synonym residual Synonym quotient ideal Related topic SumOfIdeals Related topic ProductOfIdeals Related topic Submodule Related topic ArithmeticalRing