Let $R$ be the $k$-subalgebra of $k(t)$ generated by the set $k[t]$, of all polynomials, and a pair of rational functions: ${1\over{t-1}}$ and ${1\over{t-2}}$. Is the ring $R$ a PID?
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1possible duplicate of Is the localization of a PID a PID? – Thomas Andrews Aug 21 '15 at 02:13
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Rough outline of how you might prove this: Given any ideal $I$ of $R$, consider the polynomials in the ideal, $I_1=I\cap k[t]$. Show that $I_1$ is an ideal of $k[t]$, hence principal in $k[t]$. Then show that its generator is also a generator for all of $I$ in $R$.
The basic property you'll use is that for every element $f\in R$, there is an $m$ so that $(t-1)^m(t-2)^mf\in k[t]$.
Thomas Andrews
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