Let $R$ be a commutative ring having at least one regular element and $T$ its total ring of fractions. Let $\mathfrak{a} := (a_0,\,a_1,\,\ldots,\,a_{m-1})$ and $\mathfrak{b} := (b_0,\,b_1,\,\ldots,\,b_{n-1})$ be two fractional ideals of $R$ . Then the product submodule $\mathfrak{ab}$ of $T$ is also a fractional ideal of $R$ and is generated by all the elements $a_ib_j$ , thus having a generating set of $mn$ 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

(1)

Note that the generators in (1) are the same as the coefficients in the product of the polynomials $f := a_0\!+\!a_1X\!+\cdots+\!a_{m-1}X^{m-1}$ and $g := b_0\!+\!b_1X\!+\cdots+\!b_{n-1}X^{n-1}$ . Thus we may call the fractional ideals $\mathfrak{a}$ and $\mathfrak{b}$ of $R$ the coefficient modules $\mathfrak{m}_f$ and $\mathfrak{m}_g$ of the polynomials $f$ and $g$ (they are $R$ -modules). Hence the formula (1) may be rewritten as

(2)

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 $k$ such that the $R$ -submodules of $T$ generated by the coefficients of the polynomials $f$ , $g$ and $fg$ satisfy the equality

(3)

Bibliography

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).