Let $R = \mathbb{C}[t]$ be a ring of polynomials in variable $t$ with coefficients in the field of complex numbers $\mathbb{C}$ and let $$M = R[x]/(tx-1).$$

Goal: I need to show that $$M \cong R[t^{-1}]$$ where $R[t^{-1}]$ is the localization of $R$ with respect to the multiplicative set $$S =\{t^{-i} \quad | i \in \mathbb{N}\}$$

For me to show the isomorphism, then I need to know the most natural map between $R[x]$ and $R[t^{-1}]$.

My guess is the following evaluation map $$\phi_t: R[x] \rightarrow R[t^{-1}]$$ that sends $$ f(x) \mapsto \frac{f(x)}{t}$$ Is this map good? This map won't give me the kernel $tx-1$ and thats exactly my problem

I can prove the isomorphism if I know the most natural map.


  • $\begingroup$ Hint: Map $x \to t^{-1}$. $\endgroup$
    – Ken Duna
    Apr 5, 2017 at 23:54
  • $\begingroup$ @Ken Duna Got it. Thanks. You should post that as an answer so that I can accept. Please do cos you have just saved me from stressing out lol. $\endgroup$
    – Jaynot
    Apr 6, 2017 at 0:09
  • $\begingroup$ See also math.stackexchange.com/questions/1677766/… . $\endgroup$ Apr 21, 2019 at 6:10

1 Answer 1


Per request:

Define $\varphi : R[x] \to R[t^{-1}]$ by mapping $x \mapsto t^{-1}$. Then $\text{ker}(\varphi) = (tx -1)$, and by the First Isomorphism Theorem, $$ \frac{R[x]}{(tx-1)} \cong R[t^{-1}]_.$$


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