Let be a commutative ring having at least one regular element and its total ring of fractions. Let and be two fractional ideals of (see the entry “fractional ideal of commutative ring”). Then the product submodule of is also a fractional ideal of and is generated by all the elements , thus having a generating set of elements.
Such a generating set may be condensed in the case of any Dedekind domain, especially for the fractional ideals of any algebraic number field one has the multiplication formula
Here, the number of generators is only (in principle, every ideal of a Dedekind domain has a generating system of two elements). The formula is characteristic still for a wider class of rings which may contain zero divisors, viz. for the Prüfer rings (see ), but then at least one of and must be a regular ideal.
Note that the generators in (1) are formed similarly as the coefficients in the product of the polynomials and . Thus we may call the fractional ideals and of the coefficient modules and of the polynomials and (they are -modules). Hence the formula (1) may be rewritten as
This formula says the same as Gauss’s lemma I for a unique factorization domain .
Arnold and Gilmer  have presented and proved the following generalisation of (2) which is valid under much less stringent assumptions than the ones requiring to be a Prüfer ring (initially: a Prüfer domain); the proof is somewhat simplified in .
Theorem (Dedekind–Mertens lemma). Let be a subring of a commutative ring . If and are two arbitrary polynomials in the polynomial ring , then there exists a non-negative integer such that the -submodules of generated by the coefficients of the polynomials , and satisfy the equality
- 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).