# 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 $\U0001d51e:=({a}_{0},{a}_{1},\mathrm{\dots},{a}_{m-1})$ and $\U0001d51f:=({b}_{0},{b}_{1},\mathrm{\dots},{b}_{n-1})$ be two fractional ideals^{} of $R$
(see the entry “fractional ideal of commutative ring”). Then
the product submodule $\U0001d51e\U0001d51f$ of $T$ is also a
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 of any algebraic number field^{} one has the multiplication formula

$\U0001d51e\U0001d51f=({a}_{0}{b}_{0},{a}_{0}{b}_{1}+{a}_{1}{b}_{0},{a}_{0}{b}_{2}+{a}_{1}{b}_{1}+{a}_{2}{b}_{0},\mathrm{\dots},{a}_{m-1}{b}_{n-1}).$ | (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 $\U0001d51e$ and $\U0001d51f$ must be a regular ideal.

Note that the generators in (1) are formed similarly as the coefficients^{} in the product of the polynomials^{}
$f(X):={f}_{0}+{f}_{1}X+\mathrm{\cdots}+{f}_{m-1}{X}^{m-1}$ and
$g(X):={g}_{0}+{g}_{1}X+\mathrm{\cdots}+{g}_{n-1}{X}^{n-1}$. Thus we may call the fractional ideals $\U0001d51e$ and $\U0001d51f$ of $R$ the coefficient modules ${\U0001d52a}_{f}$ and ${\U0001d52a}_{g}$ of the polynomials $f$ and $g$ (they are $R$-modules). Hence the formula (1) may be rewritten as

${\U0001d52a}_{f}{\U0001d52a}_{g}={\U0001d52a}_{fg}.$ | (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

${\U0001d52a}_{f}^{n+1}{\U0001d52a}_{g}={\U0001d52a}_{f}^{n}{\U0001d52a}_{fg}.$ | (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 |