ideal inverting in Prüfer ring

Theorem. Let $\mathfrak{a}_{1}$ , …, $\mathfrak{a}_{n}$ be invertible fractional ideals of a Prüfer ring. Then also their sum and intersection are invertible, and the inverse ideals of these are obtained by the formulae resembling de Morgan’s laws:

(\mathfrak{a}_{1}+\cdots+\mathfrak{a}_{n})^{-1}\;=\;\mathfrak{a}_{1}^{-1}\cap% \cdots\cap\mathfrak{a}_{n}^{-1}

(\mathfrak{a}_{1}\cap\cdots\cap\mathfrak{a}_{n})^{-1}\;=\;\mathfrak{a}_{1}^{-1% }+\cdots+\mathfrak{a}_{n}^{-1}

This is due to the fact, that the sum of any ideals is the smallest ideal containing these ideals and the intersection of the ideals is the largest ideal contained in each of these ideals. Cf. sum of ideals, quotient of ideals.

References

1 J. Pahikkala: “Some formulae for multiplying and inverting ideals”. $-$ Annales universitatis turkuensis 183. Turun yliopisto (University of Turku) 1982.

Title	ideal inverting in Prüfer ring
Canonical name	IdealInvertingInPruferRing
Date of creation	2015-05-06 14:34:48
Last modified on	2015-05-06 14:34:48
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	15
Author	pahio (2872)
Entry type	Theorem
Classification	msc 13C13
Related topic	DualityInMathematics
Related topic	DualityOfGudermannianAndItsInverseFunction