How can I proceed to find a maximal principal ideal in $\mathbb Z_{(2)}[x]$?
I know the answer in the sense that i know that $(2x+1)$ is a maximal principal ideal of that polynomial ring. But if i didn't, how could i reach that result, i.e. how could i reason to answer?
Since $\Bbb Z_{(2)}\subseteq \Bbb Q$ contains all fractions with odd denominator, in order to get (something isomorphic to) all of $\Bbb Q$ (i.e. a field), you would need to use $x$ to express even-denominatored fractions somehow. Dividing out by $(2x+1)$ gives $x$ the role of $-1/2$, which is sufficient, but far from the only solution.
