Geodesic
A~construction of principal ideal rings
Fundamentalʹnaâ i prikladnaâ matematika, Tome 6 (2000) no. 4, pp. 1257-1261

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Let $K$ be an algebraic number field, and let $R$ be the ring that consists of “polynomials” $a_1x^{\lambda_1}+\ldots+a_s x^{\lambda_s}$ ($a_i\in K$, $\lambda_i\in\mathbb{Q}$, $\lambda_i\geq0$). Consider the set of elements $S$ closed under multiplication and generated by the elements $x^{1/m}$, $1+x^{1/m}+\ldots+x^{k/m}$ ($m$ and $k$ vary). We prove that the ring $RS^{-1}$ is a principal ideal ring. 
@article{FPM_2000_6_4_a21,
     author = {Yu. V. Kuz'min},
     title = {A~construction of principal ideal rings},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {1257--1261},
     publisher = {mathdoc},
     volume = {6},
     number = {4},
     year = {2000},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2000_6_4_a21/}
}
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Yu. V. Kuz'min. A~construction of principal ideal rings. Fundamentalʹnaâ i prikladnaâ matematika, Tome 6 (2000) no. 4, pp. 1257-1261. http://geodesic.mathdoc.fr/item/FPM_2000_6_4_a21/