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[parent] product of finitely generated ideals (Definition)

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

$\displaystyle \mathfrak{ab} = (a_0b_0,\,a_0b_1\!+\!a_1b_0,\,a_0b_2\!+\!a_1b_1\!+\!a_2b_0,\,\ldots,\,a_{m-1}b_{n-1}).$ (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 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

$\displaystyle \mathfrak{m}_f\mathfrak{m}_g = \mathfrak{m}_{fg}.$ (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 1   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
$\displaystyle \mathfrak{m}_f^{k+1}\,\mathfrak{m}_g = \mathfrak{m}_f^k\,\mathfrak{m}_{fg}.$ (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).



"product of finitely generated ideals" is owned by pahio.
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See Also: ideal generators in Prüfer ring, ideal decomposition in Dedekind domain, entries on finitely generated ideals, unique factorization and ideals in ring of integers

Other names:  special cases of ideal product

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multiplication rule gives inverse ideal (Theorem) by pahio
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Cross-references: equality, integer, polynomial ring, subring, Prüfer domain, coefficient modules, fractional ideals, polynomials, product, coefficients, regular ideal, Prüfer rings, viz, zero divisors, contain, rings, class, ideal, generators, multiplication, algebraic number field, Dedekind domain, generating set, generated by, product submodule, total ring of fractions, regular element, commutative ring
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This is version 16 of product of finitely generated ideals, born on 2005-07-10, modified 2008-03-22.
Object id is 7217, canonical name is ProductOfFinitelyGeneratedIdeals.
Accessed 1790 times total.

Classification:
AMS MSC: 16D25 (Associative rings and algebras :: Modules, bimodules and ideals :: Ideals)
 13A15 (Commutative rings and algebras :: General commutative ring theory :: Ideals; multiplicative ideal theory)
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