Localisation is isomorphic to a quotient of polynomial ring

I am having trouble with the following problem.

Let $R$ be an integral domain, and let $a \in R$ be a non-zero element. Let $D = \{1, a, a^2, ...\}$. I need to show that $R_D \cong R[x]/(ax-1)$.

I just want a hint.

Basically, I've been looking for a surjective homomorphism from $R[x]$ to $R_D$, but everything I've tried has failed. I think the fact that $f(a)$ is a unit, where $f$ is our mapping, is relevant, but I'm not sure. Thanks

You can use the universal properties, namely, the universal property of the localization, of the polynomial ring, and of the quotient.

Recall that if $R$ is a commutative ring and $D$ is a multiplicative subset, there is a homomorphism $\phi\colon R\to R_D$, given by $\phi(r) = \frac{rd}{d}$ (where $d\in D$ is arbitrary); this map is well-defined, and has the universal property:

If $T$ is any ring, and $f\colon R\to T$ is a ring homomorphism with the property that $f(d)$ is a unit in $T$ for each $d\in D$, then there exists a unique ring homomorphism $\mathcal{F}\colon R_D\to T$ such that $\mathcal{F}\circ\phi = f$.

Consider the natural embedding $R\to R[x]$ followed by the quotient map $R[x]\to R[x]/(ax-1)$. Call this $f$. Note that $f(a)$ is a unit in $R[x]/(ax-1)$, hence so are $f(a^n)$ for every $n$. Therefore, every element of $D$ is mapped to a unit in $R[x]/(ax-1)$, which means that there is a unique homomorphism $\mathcal{F}\colon R_D\to R[x]/(ax-1)$ with the property that $\mathcal{F}\circ \phi = f$.

The claim is that $\mathcal{F}$ is an isomorphism.

To establish this, we construct an inverse. The universal property of the polynomial ring $R[x]$ tells us that if we specify how to map $R$ and what element we want $x$ to map to, we get a homomorphism. We define a homomorphism $R[x]\to R_D$ by mapping the coefficient ring $R$ to $R_D$ using $\phi$, and mapping $x$ to $\frac{1}{a}\in R_D$. This gives us a homomorphism $g\colon R[x]\to R_D$.

Now, the polynomial $ax-1$ is mapped to $0$: $g(ax-1) = \phi(a)\frac{1}{a}-\phi(1) = \frac{aa}{a}\frac{1}{a} - \frac{a}{a} = 0_{R_D}$, so by the universal property of the quotient, the map $g$ factors through $R[x]/(ax-1)$. That is, there is a unique homomorphism $\mathcal{G}\colon R[x]/(ax-1)\to R_D$ such that $g = \mathcal{G}\circ \pi$, where $\pi\colon R[x]\to R[x]/(ax-1)$ is the canonical projection.

So now we have homomorphism $\mathcal{F}\colon R_D\to R[x]/(ax-1)$ and $\mathcal{G}\colon R[x]/(ax-1)\to R_D$. I claim that $\mathcal{G}$ is the inverse of $\mathcal{F}$.

First, consider $$\begin{array}{rcccl} &&R&&\\ &{\small\phi}\swarrow & {\small f}\downarrow&\searrow{\small\phi}\\ R_D & \stackrel{\mathcal{F}}{\to} & \frac{R[x]}{(ax-1)} &\stackrel{\mathcal{G}}{\to} & R_D \end{array}$$ Now, notice that $\mathcal{G}f=\phi$, since elements of $R$ in $R[x]$ are mapped to $R_D$ as $\phi$. So this diagram commutes; that is, $\mathcal{GF}\phi =\mathrm{id}_{R_D}\phi$. But the universal property of the localization says that there is a unique map $R_D\to R_D$ that makes the diagram $$\begin{array}{rcl} &R&\\ {\small\phi}\swarrow &&\searrow{\small\phi}\\ R_D & \longrightarrow& R_D \end{array}$$ commute; clearly the identity does, but we just saw that $\mathcal{GF}$ does as well. That means that we must have $\mathcal{GF} = \mathrm{id}_{R_D}$.

On the other hand, consider the composition $\mathcal{FG}\colon R[x]/(ax-1)\to R[x]/(ax-1)$. The restriction to (the image of) $R$ of this map is just $$\mathcal{FG}(\pi(r)) = \mathcal{F}(g(r)) = \mathcal{F}(\phi(r)) = f(r) = r+(ax-1)$$ (where $(ax-1)$ means the ideal of $R$, not the element $ax-1$). And the image of the class of $x$ is $$\mathcal{FG}(\pi(x)) = \mathcal{F}(g(x)) = \mathcal{F}\left(\frac{1}{a}\right) = f(a)^{-1} = x+(ax-1).$$ So the map $\mathcal{FG}$ agrees with the identity on $\pi(R)$ and on $\pi(x)$, hence equals the identity. So $\mathcal{FG}=\mathrm{id}_{R[x]/(ax-1)}$.

Thus, $\mathcal{F}=\mathcal{G}^{-1}$, so $\mathcal{F}$ is an isomorphism.

Added. As an alternative of the latter part: once we know that $\mathcal{GF}=\mathrm{id}_{R_D}$, we conclude that $\mathcal{F}$ is one-to-one. Now notice that $\mathcal{F}$ is onto: since $\mathcal{F}\circ \phi = f$, the image of $\mathcal{F}$ contains the image of $f$; the image of $f$ includes the image of all scalars under the projection $R[x]\to R[x]/(ax-1)$. The image of $\mathcal{F}$ also includes the image of $x$, since $\mathcal{F}(\frac{1}{a}) = \mathcal{F}(a)^{-1}$, and the inverse of $\mathcal{F}(a)$ is $x+(ax-1)$. Since $x+(ax-1)$ and $r+(ax-1)$, $r\in R$, generate $R[x]/(ax-1)$, it follows that $\mathcal{F}$ is onto. Since it was already one-to-one, $\mathcal{F}$ is an isomorphism.

Here's another answer using the universal property in another way (I know it's a bit late, but is it ever too late ?)

As for universal properties in general, the ring satisfying the universal property described by Arturo Magidin in his answer is unique up to isomorphism. Thus to show that $R[x]/(ax-1) \simeq R_D$, it suffices to show that $R[x]/(ax-1)$ has the same universal property !

But that is quite easy: let $\phi: R\to T$ be a ring morphism such that $\phi(a) \in T^{\times}$.

Using the universal property of $R[x]$, we get a unique morphism $\overline{\phi}$ extending $\phi$ with $\overline{\phi}(x) = \phi(a)^{-1}$.

Quite obviously, $ax -1 \in \operatorname{Ker}\overline{\phi}$. Thus $\overline{\phi}$ factorizes uniquely through $R[x]/(ax-1)$.

Thus we get a unique morphism $\mathcal{F}: R[x]/(ax-1) \to T$ with $\mathcal{F}\circ \pi = \phi$, where $\pi$ is the canonical map $R\to R[x]/(ax-1)$. This shows that $\pi: R \to R[x]/(ax-1)$ has the universal property of the localization, thus it is isomorphic to the localization.

This is essentially another way of seeing Arturo Magidin's answer

Define $\phi: R[x] \rightarrow R_D$ by sending $x$ to $1/a$. We will prove that the kernel of $\phi$ is the ideal of $R[x]$ generated by $ax -1$.

Let $p\left(x\right) \in \ker \phi$. Note that $p(x)$ can be naturally viewed as an element of $R_D[x]$. Since $\phi(p(x))=0$, we have $p(1/a)=0$ in $R_D$. So $1/a$ is a root of $p(x) \in R_D[x]$. Since the leading coefficient of $ax-1$ is a unit in $R_D$, Euclidean division applies and we get that $p(x) = (ax-1) g(x)$ for some $g(x) \in R_D[x]$. Inspecting coefficients on both sides starting from lowest degree terms, we see that in fact $g(x) \in R[x]$ and we are done.

PS: This answer is inspired by Robert Green's answer (and our subsequent interaction) to an identical question that i recently asked, being ignorant of the existence of this post.

In the quotient $R[x]/(ax-1)$, the class of $x$ is invertible. What is its inverse?

This will tell you where $x$ must be mapped under $R[x]\to R_D$ if this must will induce an isomorphism $R[x]/(ax-1)\to R_D$.