PlanetMath: ideal generators in Prüfer ring

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ideal generators in Prüfer ring

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Let be a Prüfer ring with total ring of fractions . Let $\mathfrak{a}$ and $\mathfrak{b}$ be fractional ideals of , generated by and elements of , respectively.

Then the sum ideal $\mathfrak{a+b}$ may, of course, be generated by elements.
If $\mathfrak{a}$ or $\mathfrak{b}$ is regular, then the product ideal $\mathfrak{ab}$ may be generated by elements, since in Prüfer rings the formula
$\displaystyle (a_1, \,...,\,a_m)(b_1,\,...,\,b_n) = (a_1b_1,\,a_1b_2+a_2b_1,\,a_1b_3+a_2b_2+a_3b_1,\, ...,\,a_mb_n)$
holds.
If both $\mathfrak{a}$ and $\mathfrak{b}$ are regular ideals, then the intersection $\mathfrak{a}\cap\mathfrak{b}$ and the quotient ideal $\mathfrak{a\colon\!b} = \{r \in R\vert \quad r\mathfrak{b} \subseteq \mathfrak{a}\}$ both may be generated by elements.
If $\mathfrak{a}$ is regular, then it is also invertible. Its inverse ideal has the expression
$\displaystyle \mathfrak{a}^{-1} = [R:\mathfrak{a}] = \{t\in T\vert\quad t\mathfrak{a} \subseteq R\}$
and may be generated by elements of (see the generators of inverse ideal).

Cf. also the two-generator property.

Bibliography

J. Pahikkala: ``Some formulae for multiplying and inverting ideals''.

Annales universitatis turkuensis 183. Turun yliopisto (University of Turku) 1982.

"ideal generators in Prüfer ring" is owned by pahio.

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See Also: fractional ideal, product of finitely generated ideals

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Cross-references: two-generator property, generators of inverse ideal, regular, quotient ideal, intersection, regular ideals, ideal, generated by, sum ideal, fractional ideals, total ring of fractions, Prüfer ring
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This is version 17 of ideal generators in Prüfer ring, born on 2004-08-14, modified 2008-03-11.
Object id is 6102, canonical name is IdealGeneratorsInPruferRing.
Accessed 1345 times total.

Classification:

13C13 (Commutative rings and algebras :: Theory of modules and ideals :: Other special types)

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