Talk:Composition ring

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Discussion copied from Wikipedia talk:WikiProject Mathematics:

The article titled composition ring may be a good start, but it lacks references and has only one concrete example, and says nothing about what can be done with the concept. Can anyone add anything? Michael Hardy (talk) 22:52, 19 May 2008 (UTC)

A second example would be nice. JRSpriggs (talk) 00:30, 21 May 2008 (UTC)

For any ring R the set of all functions from R to R gives an obvious example. Also subrings closed under composition would give examples, for instance continuous, differentialble, analytic, etc. functions in case of the real or complexes (or even meromorphic functions, which is not a subring). I don't think these "functions of one variable" examples are the most interesting ones from an algebraic point of view though. Formal power series rings do not seem to work, as substitution can only be defined if the series substituted has no constant term. I know one example that I would personally consider interesting, namely the ring of symmetric functions (having infinitely many variables) with so-called plethystic substitution, which amounts infomrally speaking to writing one symmetric function as sum of monomials (with unit coefficients!) and substituting those monomials for the variables of another symmetric function. It is not easy though to define this properly in the presence of negative (or non-integer) coefficients, so I would need to write that plethysm article first… The point is that if there were any useful algebraic theory for composition rings (I did not read the article referenced), it could help to understand plethystic substitution. Marc van Leeuwen (talk) 07:37, 21 May 2008 (UTC)

I retract meromorphic functions as example, or even rational functions of one variable, since composition cannot be defined when the right factor is constant and the left factor has a pole at that value; a rare case, but enough to break the algebraic construction. By the way, I think the Math Review of the referenced paper provides more material allowing to expand the current article. I am somewhat disturbed by another paper [1] which calls composition ring certain structures (like the zero-preserving functions from a ring to itself, and also power-series examples) that obviously lack a multiplicative identity, implying a different definition of "composition ring" Marc van Leeuwen (talk) 09:14, 21 May 2008 (UTC)

There are two competing definitions of ring (mathematics); one requires multiplicative identity, the other doesn't. I think we have a convention on the subject; I forget what it is. It may be simplest to add a section, especially since it is trivial to adjoin a multiplicative identity. Septentrionalis PMAnderson 17:05, 21 May 2008 (UTC)

I know there are two definitions of ring, and one school distinguishing "rings" and "rngs", the other speaking of "unitary rings" and "rings". This is annoying enough (I'm sure the wikipedia math articles are not quite consistent in their terminology), but usually not really a problem. I've often seen the remark that it is trivial to adjoin a multiplicative identity, and I disagree it because it is not that trivial (try it without looking up a reference) nor very satifactory (try applying the construction to the rng of even integers, and see if you like the result). But here there is something more serious going on. The references I saw refer to "ring" without specifying which brand they mean, so it is hard to figure out what they are talking about. But the definition at composition ring clearly supposes a multiplicative identity, and even gives an axiom involving it. Whereas the non-examples I mentioned (formal power series and rational function rings) show that the presence of (non-zero) constants form a serious obstruction to defining composition: both formal power series without constant term and zero-preserving rational functions (which in particular have no pole at 0) can be made into composition rngs (without multiplicative unit). Also I am convinced that it is impossible to adjoin a multiplicative unit to these examples (the usual construction might give a ring, but without composition operation defined). I am inclined to believe we should take out the multiplicative unit in the composition ring article (and this discussion should maybe be moved to its talk page). Marc van Leeuwen (talk) 09:18, 22 May 2008 (UTC)

• The sources (e.g. Jacobson) say it is trivial, and it is (take the product with Z, with multiplication (m,r)(n, s)= (mn, ms + nr + rs) where m, n are integers; r, s in the ring R) which becomes obvious if written additively.
• I don't see why either of these rng examples doesn't have the identity function as composition identity; the article should explain.
• In any case, why can't those examples just become another section with heading "Composition rngs"? Septentrionalis PMAnderson 23:32, 22 May 2008 (UTC)

Composition in polynomial rings

The current article says: The polynomial ring R[X] is a composition ring where $(f\circ g)(x)=f(g(x))$ for all $f,g\in R$.

However, an element of the ring R[X] isn't necessarily a mapping. It's something more abstract. So the concept of f(g(x)) requires a definition different than the definition that applies to the composition of functions. However, I like the brevity of this article. The technical details could be handled by defining f(g(x)) in the article on polynomial rings.

Tashiro~enwiki (talk) 15:19, 30 October 2019 (UTC)