cancellation ideal

Let R be a commutative ring containing regular elementsPlanetmathPlanetmath and 𝔖 be the multiplicative semigroup of the non-zero fractional idealsMathworldPlanetmathPlanetmath of R.  A fractional ideal π”ž of R is called a cancellation ideal or simply cancellative, if it is a cancellative element of 𝔖, i.e. if

π”žβ’π”Ÿ=π”žβ’π” β‡’π”Ÿ=π” β€ƒβˆ€π”Ÿ,π” βˆˆπ”–.
  • β€’

    Each invertible ideal is cancellative.

  • β€’

    A finite product π”ž1β’π”ž2β’β€¦β’π”žm of fractional ideals is cancellative iff every π”ži is such.

  • β€’

    The fractional ideal  π”ž/r:={a⁒r-1:aβˆˆπ”ž},  where π”ž is an integral ideal of R and r a regular element of R, is cancellative if and only if π”ž is cancellative in the multiplicative semigroup of the non-zero integral ideals of R.

  • β€’

    If  r∈R,  then the principal idealMathworldPlanetmathPlanetmathPlanetmathPlanetmath (r) of R is cancellative if and only if r is a regular element of the total ring of fractionsMathworldPlanetmath of R.

  • β€’

    If  π”ž1+π”ž2+…+π”žm  is a cancellation ideal and n a positive integer, then


    In particular, if the ideal  (a1,a2,…,am)  of R is cancellative, then



  • 1 R. Gilmer: Multiplicative ideal theory.  Queens University Press. Kingston, Ontario (1968).
  • 2 M. Larsen & P. McCarthy: Multiplicative theory of ideals.  Academic Press. New York (1971).
Title cancellation ideal
Canonical name CancellationIdeal
Date of creation 2015-05-06 14:49:08
Last modified on 2015-05-06 14:49:08
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 11
Author pahio (2872)
Entry type Definition
Classification msc 13B30
Synonym cancellative ideal
Related topic CancellativeSemigroup
Related topic IdealDecompositionInDedekindDomain
Defines cancellative