A construction of principal ideal rings

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.