product of finitely generated ideals

Let R be a commutative ring having at least one regular elementPlanetmathPlanetmath and T its total ring of fractionsMathworldPlanetmath.  Let  𝔞:=(a0,a1,,am-1)  and  𝔟:=(b0,b1,,bn-1)  be two fractional idealsMathworldPlanetmathPlanetmath of R (see the entry “fractional ideal of commutative ring”).  Then the product submodule𝔞𝔟  of T is also a of R and is generated by all the elements aibj, thus having a generating set of  mn  elements.

Such a generating set may be condensed in the case of any Dedekind domainMathworldPlanetmath, especially for the of any algebraic number fieldMathworldPlanetmath one has the multiplication formula

𝔞𝔟=(a0b0,a0b1+a1b0,a0b2+a1b1+a2b0,,am-1bn-1). (1)

Here, the number of generatorsPlanetmathPlanetmath is only  m+n-1 (in principle, every ideal of a Dedekind domain has a generating system of two elements (  The formula is characteristicPlanetmathPlanetmath ( still for a wider class of rings R which may contain zero divisors, viz. for the Prüfer rings (see [1]), but then at least one of 𝔞 and 𝔟 must be a regular ideal.

Note that the generators in (1) are formed similarly as the coefficientsMathworldPlanetmath in the product of the polynomialsMathworldPlanetmathPlanetmathf(X):=f0+f1X++fm-1Xm-1  and  g(X):=g0+g1X++gn-1Xn-1.  Thus we may call the fractional ideals 𝔞 and 𝔟 of R the coefficient modules 𝔪f and 𝔪g of the polynomials f and g (they are R-modules).  Hence the formula (1) may be rewritten as

𝔪f𝔪g=𝔪fg. (2)

This formula says the same as Gauss’s lemma I for a unique factorization domainMathworldPlanetmath R.

Arnold and Gilmer [2] have presented and proved the following generalisation of (2) which is valid under much less stringent assumptions than the ones requiring R to be a Prüfer ring (initially: a Prüfer domain); the proof is somewhat simplified in [1].

Theorem (Dedekind–Mertens lemma).   Let R be a subring of a commutative ring T.  If f and g are two arbitrary polynomials in the polynomial ring T[X], then there exists a non-negative integer n such that the R-submodules of T generated by the coefficients of the polynomials f, g and fg satisfy the equality

𝔪fn+1𝔪g=𝔪fn𝔪fg. (3)


  • 1 J. Pahikkala: “Some formulae for multiplying and inverting ideals”.  – Ann. Univ. Turkuensis 183 (A) (1982).
  • 2 J. Arnold & R. Gilmer: “On the contents of polynomials”.  – Proc. Amer. Math. Soc. 24 (1970).
Title product of finitely generatedMathworldPlanetmathPlanetmath ideals
Canonical name ProductOfFinitelyGeneratedIdeals
Date of creation 2015-05-05 19:19:39
Last modified on 2015-05-05 19:19:39
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 29
Author pahio (2872)
Entry type Definition
Classification msc 13A15
Classification msc 16D25
Classification msc 16D10
Synonym special cases of ideal product
Related topic PruferRing
Related topic IdealGeneratorsInPruferRing
Related topic IdealDecompositionInDedekindDomain
Related topic EntriesOnFinitelyGeneratedIdeals
Related topic UniqueFactorizationAndIdealsInRingOfIntegers
Related topic ContentOfAPolynomial
Related topic WellDefinednessOfProductOfFinitelyGeneratedIdeals
Defines Dedekind–Mertens lemma