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.
$\\mathfrak{a}\\subseteq\\mathfrak{b}\\,\\,\\,\\,\\Rightarrow\\,\\,\\,\\mathfrak{a}:%\\mathfrak{c}\\subseteq\\mathfrak{b}:\\mathfrak{c}\\,\\,\\land\\,\\,\\mathfrak{c}:%\\mathfrak{a}\\supseteq\\mathfrak{c}:\\mathfrak{b}$
2.
$\\mathfrak{a}:(\\mathfrak{b}\\mathfrak{c})=(\\mathfrak{a}:\\mathfrak{b}):\\mathfrak{c}$
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 (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