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.
Thanks.
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, $$ \frac{R[x]}{(tx-1)} \cong R[t^{-1}]_.$$