quotient of ideals

Let R be a commutative ring having regular elementsPlanetmathPlanetmath and let T be its total ring of fractionsMathworldPlanetmath.  If π”ž and π”Ÿ are fractional idealsPlanetmathPlanetmath of R, then one can define two different or residuals of π”ž by π”Ÿ:

  • β€’

    π”ž:π”Ÿ:={r∈R| rπ”ŸβŠ†π”ž}

  • β€’

    [π”ž:π”Ÿ]:={t∈T| tπ”ŸβŠ†π”ž}

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


In the special case that R has non-zero unity and π”Ÿ has the inverse ideal π”Ÿ-1, we have


in particular


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

  1. 1.

    π”žβŠ†π”Ÿβ‡’π”ž:π” βŠ†π”Ÿ:π” βˆ§π” :π”žβŠ‡π” :π”Ÿ

  2. 2.

    π”ž:(π”Ÿπ” )=(π”ž:π”Ÿ):𝔠

  3. 3.


  4. 4.


Remark.  In a PrΓΌfer 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

π”ž:(π”Ÿβˆ©π” )=(π”ž:π”Ÿ)+(π”ž:𝔠)


of the two last rules if the are finitely generatedMathworldPlanetmathPlanetmath.

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 SubmoduleMathworldPlanetmath
Related topic ArithmeticalRing