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 $R$ (see the entry “fractional ideal of commutative ring”). Then the product submodule $\mathfrak{ab}$ of $T$ is also a fractional ideal of $R$ and is generated by all the elements $a_{i}b_{j}$, 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

$\displaystyle\mathfrak{ab}=(a_{0}b_{0},\,a_{0}b_{1}\!+\!a_{1}b_{0},\,a_{0}b_{2% }\!+\!a_{1}b_{1}\!+\!a_{2}b_{0},\,\ldots,\,a_{m-1}b_{n-1}).$ | (1) |

Here, the number of generators is only $m\!+\!n\!-\!1$ (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 $R$ which may contain zero divisors, viz. for the Prüfer rings (see [1]), but then at least one of $\mathfrak{a}$ and $\mathfrak{b}$ must be a regular ideal.

Note that the generators in (1) are formed similarly as the coefficients in the product of the polynomials $f(X):=f_{0}\!+\!f_{1}X\!+\cdots+\!f_{m-1}X^{m-1}$ and $g(X):=g_{0}\!+\!g_{1}X\!+\cdots+\!g_{n-1}X^{n-1}$. 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

$\displaystyle\mathfrak{m}_{f}\mathfrak{m}_{g}=\mathfrak{m}_{fg}.$ | (2) |

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

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

$\displaystyle\mathfrak{m}_{f}^{n+1}\,\mathfrak{m}_{g}=\mathfrak{m}_{f}^{n}\,% \mathfrak{m}_{fg}.$ | (3) |

## References

- 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).