[1] Bergman, G. The diamond lemma for ring theory,
Advances in Mathematics 29 (1978) 178-218
http://dx.doi.org/10.1016/0001-8708(78)90010-5
[2] Bergman, G. An Invitation to General Algebra and Universal Constructions.
http://math.berkeley.edu/~gbergman/245/

Next we show that the standard sum-of-monomials representation yields a normal form for elements of T/Q, i.e. every element of T/Q is uniquely represented by some such normalized polynomial of T. Existence is trivial: simply apply the ring axioms to reduce a representative to normal polynomial form. It's less trivial to prove uniqueness, i.e. that distinct normal forms represent distinct elements of T/Q. For this there is a common trick that often succeeds: employ a convenient representation of the ring. Here a regular representation does the trick. This method is called the "van der Waerden trick" since he employed it in constructions of group coproducts (1948) and Clifford algebras (1966).

Notice that this development is pleasingly conceptual: R[x] is constructed quite naturally as the solution to a universal mapping problem - a problem which is motivated by the desire to be able to perform generic proofs as in said proofs of det identities. Everything is well-motivated - nothing is pulled out of a hat.

The same construction of free algebras works much more generally, e.g. for any class of algebras that admit a first-order equational axiomatization. Although there are also a few other known methods to construct such free algebras, this method is the most natural pedagogically and constructively. Indeed, this is the way most computer algebra systems implement free algebras. The difficulty lies not so much in the construction of the free algebra but, rather, in inventing normal-form algorithms so that one may compute effectively in such free algebras. Although this is trivial for rings and groups, for other algebras it can be quite difficult - e.g. the free modular lattice on 5 generators has undecidable word problem, i.e. no algorithm exists for deciding equality. Of course much work has been done trying to discover such normal form algorithms, e.g. google Knuth-Bendix completion, Bergman's diamond lemma. Ditto for algorithms for computing in quotients of free algebras, i.e. algebras presented by generators and relations, e.g. Grobner bases, Todd-Coxeter, etc.

I actually once thought about writing an entry on the Tricky on this technique, but laziness has prevented me to do so.

(1) As I say, when R is an infinite integral domain, the associativity of the product in R[t] follows from the injectivity of the evaluation map R[t] -> Maps(R,R), the latter being endowed with pointwise sum and product.

(2) In the general case, let Z_R be polynomial ring over Z with indeterminates {t_r | r in R}, an infinite integral domain. By Case 1, the product in Z_R[t] is associative. Since there is a natural surjective homomorphism from Z_R[t] to R[t] (if you like, you can think of this in terms composing a finitely nonzero function from N to Z_R with the natural homomorphism Z_R -> R to get a finitely nonzero function from N to R). Since the multiplication is associative on Z_R[t], it must be associative on the quotient R[t].

This is a sort of polynomial universality argument, although perhaps not the one Mr. Dubuque has in mind. (I am interested to see that argument.) It is also possibly too slick for its own good: as I explain in my notes, one can simply verify, once and for all, that the convolution product in a semigroup algebra is associative and regard this as being a "fundamental instance of associativity" like the associativity of composition of functions. However, it is always nice to see more than one approach.

To your list:

1. I thought that motivation very scant and feel that you are reading in to it more than is there, perhaps because this is an issue that you have encountered time and time again. (I say that to encourage you to post your answer as an answer to your own question rather than to this one.)

2. "Perhaps". Without the participation of the questioner, speculation is all we have.

3. Nothing stops the OP commenting, or editing the question to try to make it better.

4. When a question can admit several different levels of answer, only one will generally be at the right level for the OP. Since MO is about helping people, it's useful to know enough information to help select the right level. I've gone on at length about this elsewhere on meta so am hesitant at repeating myself (actually, I'm considering starting a blog wherein I collect all my "stock answers" such as "good answers do not make good questions" so that I can just link to that in cases like this rather than have to trawl back through my old posts here). Answers to questions should primarily try to help the person who originally asked it, otherwise there's a high risk that the effort that went in to answering the question will be wasted. That effort could so easily be directed to somewhere more useful.

I agree that none of these on it's own is really sufficient grounds for closure. But taken together, and throwing in the OP's lack of participation, and I think that there is enough.

But let me end by saying that I'm really looking forward to reading this answer, wherever it ends up, and am glad that this discussion arose since I would almost certainly have missed it otherwise.

@Wadim: Hmm, you're right. I'd better make sure that I have a good cushion. I'll get a couple more of my sockpuppets to pose easy differential topology questions.

As for the points you raised:

1) The OP does explicitly reveal some motivation, namely he seeks to understand how to construct R[x] set-theoretically and to better understand the algebraic conception of polynomial rings. Such issues are not only of interest to students. For example, Pete L. Clark's answer refers to his notes on commutative algebra - where he discusses such topics at much greater length than do most algebra textbooks. There, while discussing various constructions of R[x], he remarks:

However it is tedious to verify associativity. Over the years I have developed a slogan: if you are working hard to show that some binary operation is associative, you are missing part of a bigger picture. Unfortunately this is not such a great test case for the slogan: I do not know of a truly snappy conceptual proof of the associativity of multiplication in a polynomial ring. -- Pete L. Clark, Commutative Algebra, Sec. 4.3, p. 38

In fact there is a "bigger picture", including a construction that achieves what he seeks. Such topics are probably not well-known to those who have not studied universal algebra. But they certainly deserve to be better known due to the fact that they provide deeper conceptual and foundational insight.

2) Perhaps the OP has consulted standard sources. But there is no nontrivial discussion of such in most standard algebra textbooks. Indeed, there is only one textbook on such that I think worthy of recommendation and it is not well-known.

3) Perhaps there are no comments from the OP because the thread was closed so quickly.

4) Why should the OP have to explicitly specify what constitutes a satisfactory answer? While I agree that doing so can prove very helpful, I don't agree that the lack of such should be grounds for closure.

Something should be changed if we really wish to see and do maths.

but perhaps I wouldn't put it quite so strongly. I think that MO is in fairly robust health, these days, though I do find myself wondering a bit when my recent reputation has come from this answer and this answer. I'm tempted to declare that once I get to the magic 10k that all my answers will be CW.

A good answer does not make a good question.

Why is this a bad question?

1. There is no motivation. Why is the person interested?
2. There is no indication of having made any attempt to find out from traditional sources. Why should I do this work for them?
3. There is no indication that the questioner actually cares about the answer. Despite the slew of comments and the back-and-forth, I can't find any more from the actual questioner.
4. There is no indication of what would make a satisfactory answer to the questioner.

I completely agree that there is a good question that would evoke the same answers. But why not wait until that question appears before putting in the effort of writing the answer? Wait until someone actually cares enough to read what you've written!

If, as may be the case here, someone thinks that there is an issue that is frequently overlooked even by research mathematicians, then there's a case to be made for asking-and-answering a question about it. But then write the question yourself and make it clear that here is a key issue that is frequently misunderstood. That way, you'll stand a better chance of reaching the people that you want to reach.

I must have missed that question first time round, possibly it got closed before I saw it. But if it had been left open and no fuss made about it, I would have skimmed it and not bothered reading any of the answers because it seems to be just asking "How do you define R[x]?". So I would have missed any great answers left, and given that I'm trying to understand universal algebra and stuff like that, something explaining the universality of polynomials would be something I'd be very interested in reading. So don't hide your light under a bushel! If you've got something worth saying, say it somewhere that people will hear it!

The question about the variables in polynomial rings is not of interest as stated to mathematicians. Bill is right that it could be edited into an interested question about «discussion of R[x] as a free R-algebra, universality of polynomial identities, ways to construct free algebras etc - some of which deserve to be much better known to non-algebraists (e.g analysts)», but it could also be edited into an interesting question about how to make tartiflette! (And that would turn it into a more interesting question for at least one mathematician!)

Closing a question does not stop anyone from editing it into a more sensible one. It does stop people from answering it---and answers is not the place to fix a question!

Can you, please, explain which paragraphs in FAQ does 15447 violates, according to you?

I completely agree with the sentiment that consistency in closing questions is important. The best course of action is to decide according to the stated policies. In my estimation, 33865 clearly violates the FAQ and 15447 does not. You should also keep in mind that many people able to vote on closing questions now were not in this position at the time the question was asked, and if they did but the question accumulated fewer than 5 closing votes then you wouldn't know it.

[Edited to correspond to Bill's updated initial comment]