Show existence of surjective homomorphism from ring of polynomials to ring of fractions with $\varphi(x)=1/t$

Question

Let $ R $ be an integral domain, $ t\in R\backslash\{0\} $ and $ S=\{t^n:n\ge 0 \} $. Show that there exists a surjective ring homomorphism $ \varphi:R[x]\to S^{-1}R $ that satisfies $ \varphi(x)=\dfrac{1}{t} $.

I can find a surjective function $\varphi$ with $\varphi(x)=1/t$, but it is not a ring homomorphism. But anyway, I don't really need do find $\varphi$, I only need to prove its existence. I don't even know where to start. Can someone give me a hint?

Answer 1

$\renewcommand{\phi}{\varphi}$By the universal property of polynomial rings, there is a unique ring homomorphism $\phi : R[x] \to S^{-1} R$ such that $$ \begin{cases} \phi(r) = \dfrac{r}{1} & \text{for $r \in R$},\\ \\ \phi(x) = \dfrac{1}{t}.\\ \end{cases} $$ Clearly for $r \in R$ and $n \ge 0$ $$ \phi(r x^{n}) = \phi(r) \cdot \phi(x)^{n} = \frac{r}{t^{n}}, $$ hence $\phi$ is surjective.