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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

5 Answers 5

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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.

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    Is it correct just to say that since $\mathcal{G}\circ f=\phi$ and $\mathcal{F}\circ \phi=f$, it follows that $\mathcal{G}\circ\mathcal{F}\circ \phi=\phi$ and $\mathcal{F}\circ\mathcal{G}\circ f=f$, and the surjectivity of both $\phi$ and $f$ implies $\mathcal{F}\circ\mathcal{G}=\operatorname{id}$ and $\mathcal{G}\circ\mathcal{F}=\operatorname{id}$?2017-01-11
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    Excellent! Dummit and Foote said "It is not difficult to see that." (Example (2) on page 708 in _Abstract Algebra_) You are so good at applying these universal properties. Do you know which books I can find a similar proof?2018-12-04
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    Note that this nice proof uses neither that $R$ is an integral domain (any commutative ring would do), nor that $a$ is nonzero.2018-12-07
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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 $R[x]/(ax-1)\to R_D$.

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    The only nontrivial part of the problem is *proving* that the kernel is $\:(ax-1),\:$ which proves tricky for some.2012-05-31
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    @Bill, the question asks for a hint about how to see what the map should be. You are putting the carriage before the horses, as far as I can see...2012-06-01
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    The question asks for a hint for a *proof of the isomorphism.* If finding the map proves difficult, then it will probably be a bigger hurdle to verify the kernel *equality* (the necessity of which is often overlooked by students). Hence my emphasis.2012-06-01
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    Dear Bill, I suggest you write the answer you have in mind.2012-06-01
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    See my hint in comments to Don's answer.2012-06-01
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    Well, that is also not writing the answer you have in mind...2012-06-01
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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.

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    Could you elaborate on what you mean by "inspecting coefficients on both sides starting from lowest degree terms, we see that in fact $g(x) \in R[x]$"? When I factor out $(ax-1)$ in $R_D[x]$, the lowest-degree terms are too messy to comprehend. If I start with $c_nx^n + c_{n-1}x^{n-1} + \dotsb + c_1x + c_0$, then my coefficients on $g$ are $(c_n/a)x^{n-1} + ((c_{n-1} + c_n/a)/a)x^{n-2}$, and progressively more complicated after that.2015-10-29
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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

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    Note that this nice proof uses neither that $R$ is an integral domain (any commutative ring would do), nor that $a$ is nonzero.2018-12-07
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As $\,\,ax-1=0\,\,$ in the quotient $\,\,R[x]/(-1+ax)\,\,$, there doesn't seem to be much choice here: the map $$x\to\frac{1}{a}\,\,,\,\,f(x)\to f\left(\frac{1}{a}\right)\,,\,f(x)\in R[x]$$

seems like a reasonable choice: it is almost trivial that it is a ring homom. (remember the usual evaluation map), it also is easy to show it is onto $\,R_D\,$ , and its kernel is the ideal it must be.

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    *Proving* that the kernel is $(ax-1)$ can be tricky. Did you try it?2012-05-31
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    Not really, but seeing that $\,R\,$ is an integral domain, I think we can *slip* into is fractions field and then the proof there goes as usual: a pol. vanishes at some element iff it is divided by the minimal polynomial of that element, which is either $\,ax-1\,$, if there's no demmand of the min. pol. to be monic, or its associate element (in the pol. ring) $\,x-1/a\,$2012-05-31
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    That could be circular, since in many presentations this is a precursor to constructing the fraction field (by localization). It can be done without any knowledge of fraction fields.2012-06-01
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    How'd you do it? In may case, I'd go by constructing the fractions field in a more elementary way and not as the localization wrt the prime ideal $\,\{0\}\,$ , namely: as the set of equivalence classes of elements of the form $\,a/b\,\,,\,a,b\in R\,\,,\,b\neq 0\,$ under the well-known equiv. relation $\,a/b ~ c/d \Longleftrightarrow ad=bc\,\,in\,\,R$ and the usual operations (this is the way I first learned this, way before I had a commutative algebra course).2012-06-01
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    **Hint** $\ $ Since the image is a domain, the kernel $\rm\:P\:$ is prime. Note $\rm\ g = ax\!-\!1\in P.\:$ Suppose $\rm\: f \in P.\:$ Hence $\rm\: f + f_0\:\! g =: x\:\!h \in P,\ x \not\in P\Rightarrow\ h\in P.\:$ By induction on degree $\rm\:g\:|\:h\ $ so $\rm\ g\:|\:xh-f_0g = f.\:$ Therefore $\rm\: P = (g) = (ax\!-\!1).\ $ **QED** $\ \ $2012-06-01
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    Ok, I'm almost there, but I still can't see why you can assume we can factor out $\,x\,$ in the expression for $\,f+f_0g\,$...? As far as I see, that expression could have a non-zero free doefficient. I can't also see why $\,g\mid h\,$...2012-06-01
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    **Hint** $\ $ Recall $\rm\:x\:|\:p(x)\iff p(0) = p_0 = 0.\:$ For $\rm\:p = f + f_0 g\:$ we have $\rm\:p_0 = f_0 + f_0 g_0 = f_0 - f_0 = 0.\:$ Thus $\rm\:x\:|\:p,\:$ so $\rm\:p = x\:g,\:$ for some $\rm\:g\in R[x].\:$ Finally $\rm\:g\:|\:h\:$ follows from the (implicit) induction hypothesis, namely that $\rm\:g\:$ divides all elements of $\rm\:P\:$ of smaller degree than $\rm\:f.$2012-06-01
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    Oh, wait: I didn't know $\,f_0\,$ mean $\,f(0)\,$ ! Ok, so now it is clear. Why from this $\,g\mid h\,$ though?2012-06-01
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    Since $\rm\:h\in P\:$ has smaller degree than $\rm\:f,\:$ our induction hypothesis yields that $\rm\:g\:|\:h.\:$ We are proving by induction on degree that every element $\rm\:f\in P\:$ is divisible by $\rm\:h = ax\!-\!1\in P.\:$2012-06-01
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    Oh, thank you. It's nice, yet I think going into the fractions fields is more elementary and easy, but it's always nice to learn new approaches.2012-06-01
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    Yes, but only if one already has available a rigorous development of fraction fields. Often this is not done *rigorously* until fraction fields are derived as a special case of localizations. In that case it could be circular to employ fraction fields in proofs about localizations.2012-06-01