Definition talk:Polynomial Ring

I am tempted to define a polynomial ring over $R$ (comm, with 1) as a triple $(S, f, X)$ where $S$ is a ring, $X\in S$ and $f:R\to S$ a ring homomorphism such that they satisfy the universal property.

This is the only proper way to do it, because one should not lose track of the embedding $f$ (compare with tensor product etc), and specifying $X$ allows to talk about "variable" (which is what we do all the time) in a 100% rigorous way. Conversely, $f$ and $X$ give all information that's needed.

Of course, this while keeping all existing definitions. It has to stay easy to understand. The definitions are coherent with this one, it's only the notation that needs to be changed a bit: $R[X]$ becomes $(R[X], f, X)$, where the definitions of $f$ and $X$ depend on the definition of $R[X]$ that's used (sequences/functions on monoid, etc).

Sadly, this is not backed up by literature. --barto (talk) 17:27, 18 October 2017 (EDT)

I have the impression that this is another example of a situation where the average writer allows for the convenience of not being concerned with explaining how their definition is "obviously equivalent" to that of other writers. Which makes achieving coherent coverage an absolute horror. — Lord_Farin (talk) 17:37, 18 October 2017 (EDT)
I just checked Bourbaki, hoping to get some inspiration there. What I did find is that they define "variable" as the image of something abstract in the polynomial ring, as I suggested above. But they too give only one construction so that they see no need to specify the embedding. --barto (talk) 17:56, 18 October 2017 (EDT)
Actually Bourbaki is a bit informal when defining things like "coefficients" and degree". Basically, I think this is a place where ProofWiki can take a leading role and rise above all those books. --barto (talk) 18:00, 18 October 2017 (EDT)
Okay so anyone mind if I implement the $(S, f, X)$ thing? --barto (talk) 02:59, 21 October 2017 (EDT)
Might as well. If you can try and bring the other definitions in line with that one, so as not to alienate and confuse people who are familiar with other ways to define such an object. --prime mover (talk) 04:32, 21 October 2017 (EDT)
Yup, that's the idea. --barto (talk) 05:21, 21 October 2017 (EDT)
Started. Will take some time. --barto (talk) 06:45, 21 October 2017 (EDT)
Will need to write a bit about algebras and pointed algebras first. --barto (talk) 06:56, 21 October 2017 (EDT)

Starts taking shape... I'm doing my best not to obscure everything with terminology about algebras. Not sure if we want to keep the definition by UP (introduced by me). --barto (talk) (contribs) 14:08, 4 November 2017 (EDT)

Done

The definition of pol. ring is now fixed, up to some details, and includes all previous definitions. Next steps are (1) go trough Whitelaw and his "polynomials in elements" (see Definition:Polynomial in Ring Element) and (2) heal defs like monomial, coefficient, degree, in that order. --barto (talk) (contribs) 14:11, 19 November 2017 (EST)