ideal generators in Prüfer ring

Let $R$ be a Prüfer ring with total ring of fractions $T$ . Let $\\mathfrak{a}$ and $\\mathfrak{b}$ be fractional ideals of $R$ , generated by (http://planetmath.org/IdealGeneratedByASet) $m$ and $n$ elements of $T$ , respectively.

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Then the sum ideal $\\mathfrak{a+b}$ may, of course, be generated by $m+n$ elements.
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If $\\mathfrak{a}$ or $\\mathfrak{b}$ is regular (http://planetmath.org/FractionalIdealOfCommutativeRing), then the product (http://planetmath.org/ProductOfIdeals) ideal $\\mathfrak{ab}$ may be generated by $m+n-1$ elements, since in Prüfer rings the
$(a_{1},\\,...,\\,a_{m})(b_{1},\\,...,\\,b_{n})=(a_{1}b_{1},\\,a_{1}b_{2}+a_{2}b_{1}%,\\,a_{1}b_{3}+a_{2}b_{2}+a_{3}b_{1},\\,...,\\,a_{m}b_{n})$
holds.
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If both $\\mathfrak{a}$ and $\\mathfrak{b}$ are regular ideals, then the intersection $\\mathfrak{a}\\cap\\mathfrak{b}$ and the quotient ideal $\\mathfrak{a\\colon\\!b}=\\{r\\in R|\\quad r\\mathfrak{b}\\subseteq\\mathfrak{a}\\}$ both may be generated by $m+n$ elements.
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If $\\mathfrak{a}$ is regular, then it is also invertible (http://planetmath.org/InvertibleIdeal). Its ideal has the expression (http://planetmath.org/QuotientOfIdeals)
$\\mathfrak{a}^{-1}=[R:\\mathfrak{a}]=\\{t\\in T|\\quad t\\mathfrak{a}\\subseteq R\\}$
and may be generated by $m$ elements of $T$ (see the generators of inverse ideal).

Cf. also the two-generator property.

References

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