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. &mdash; Lord_Farin (talk) 17:37, 18 October 2017 (EDT)