product of finitely generated ideals

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 of $R$ and is generated by all the elements $a_{i}b_{j}$ , thus having a generating set of $m n$ elements.

Such a generating set may be condensed in the case of any Dedekind domain, especially for the 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 (http://planetmath.org/TwoGeneratorProperty)). The formula is characteristic (http://planetmath.org/Characterization) 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 $f g$ 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).

Title	product of finitely generated 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