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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.

Turquoise Tilt
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    Where should $x$ map to in the isomorphism $A[x]/(1-ax)\to S^{-1}A$? There is a very clear and canonical choice. – daruma Jun 07 '21 at 17:31
  • @daruma Now I understand. If I define the morphism from $A[x]$ to $S^{-1}A$ and "luckily" the Kernel of the morphism is $(1-ax)$ do you think it's a "cleaner" solution or is just a waste of time? Thank you – Turquoise Tilt Jun 07 '21 at 17:50
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    I think you get the idea. I hope it is clear that $x\mapsto 1/a$ this way. – daruma Jun 07 '21 at 17:52
  • yea, clear as day. I was thinking to the Kernel so this way you shouldn't be annoyed by verifying things are well defined etc @daruma – Turquoise Tilt Jun 07 '21 at 17:55
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    Please use the search feature. This question has been asked many, many times before: 1, 2, 3, 4, 5 – Viktor Vaughn Jun 07 '21 at 19:57

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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

Sergio GGG
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