ideal generators in Prüfer ring

Let R be a Prüfer ring with total ring of fractionsMathworldPlanetmath T.  Let 𝔞 and 𝔟 be fractional idealsMathworldPlanetmathPlanetmath of R, generated by ( m and n elements of T, respectively.

  • Then the sum ideal 𝔞+𝔟 may, of course, be generated by m+n elements.

  • If 𝔞 or 𝔟 is regular (, then the product ( ideal 𝔞𝔟 may be generated by m+n-1 elements, since in Prüfer rings the



  • If both 𝔞 and 𝔟 are regular ideals, then the intersection 𝔞𝔟 and the quotient ideal𝔞:𝔟={rR|r𝔟𝔞}  both may be generated by m+n elements.

  • If 𝔞 is regular,  then it is also invertible (  Its ideal has the expression (


    and may be generated by m elements of  T (see the generators of inverse ideal).

Cf. also the two-generator property.


J. Pahikkala:  “Some formulae for multiplying and inverting ideals”. - Annales universitatis turkuensis 183.  Turun yliopisto (University of Turku) 1982.

Title ideal generatorsPlanetmathPlanetmathPlanetmath in Prüfer ring
Canonical name IdealGeneratorsInPruferRing
Date of creation 2013-03-22 14:33:04
Last modified on 2013-03-22 14:33:04
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 20
Author pahio (2872)
Entry type Result
Classification msc 13C13
Related topic FractionalIdeal
Related topic ProductOfFinitelyGeneratedIdeals