Let $A$ be a unit commutative ring and $S:= \{a^n | a \in A, n\geq 0\}$. With $S^{-1}A$ the localization of $A$ over $S$.
I'm having some problems on proving that $$S^{-1}A \cong A[X]/(1-ax)$$ I'm not searching for a complete solution maybe just some hints, thank you very much.
Just an idea: Let $b\in A$, then note that $bx^{i}=b(a^{k}x^{i+k})$. This would let you to the map that you want