Rings of Functions (Pt. II)
Point of Post: This is a continuation of this post.
Differentiable Like Subrings
Of course my statement that (most basic) analysis could in some sense be described as studying subrings of for certain spaces is hinting at one particular group of subrings in particular, subrings of differentiable ‘type’ functions. Here we mention only two particular examples. Namely, we mention the ring of -times differentiable functions for an open subset and the set of holomorphic functions on some open subset .
While the purpose of this section is only to introduce the notations for these structures I can’t help but note the following theorem from basic complex analysis. In particular, have you ever wondered why an open connected subset of is called a ‘domain’? Well, perhaps the following might elucidate things:
Theorem: Let be non-empty and open. Then, is connected if and only if is an integral domain.
Proof: Recall from basic complex analysis that if is non-zero where is open and connected then has no accumulation points and thus it’s countable (this follows from the fact that every uncountable subset of a Lindelof space has an accumulation point). Thus, if are non-zero then is countable and thus not all of .
Conversely, if is not disconnected then we can write where are non-empty open subsets of . It’s easy to see then that (the characteristic functions) are holomorphic and non-zero yet .
Polynomials as Functions
As was mentioned in our motivation for polynomial rings there is a very real distinction between a polynomial and the obvious function it represents. The example we gave was that as polynomials we have that yet as functions as is easily checked. That said, there is a natural map which takes each polynomial to its associated function. It’s easy to check that the definition of is precisely such that is a homomorphism. The question then is, when is it an injection so that we may identify with ? Well, as shown above we clearly ‘always’ can’t. In fact, it’s fairly clear that if is finite then can never be injective since is infinite and is not.
So, when can we legitimately think of as being a subring of via the above identification? I mean, obviously this is a desirable trait. In fact, it is so natural and done so freely in non-rigorous mathematics that most of us would be caught off guard (in our dilletantish days of course) by the subtle need for the distinction. Luckily, the identification is harmless in most of the cases one is most apt to make it, most notably for or . So, what is a sufficient condition on a ring for which this identification is ‘ok’? Rewording this, we really want to know when is injective, or when . Taking it one step further we come to the simplest interpretation of this problem: under what conditions on a ring does for all implies ? Taking our ‘known’ (more correctly stated ‘presumed’) cases of and one may be inclined to guess that a sufficient condition is that is an (infinite) field. Well, while this is correct it leaves out a lot of very natural rings such as and, thinking ahead, (for some field (not necessarily infinite) ). Luckily, both of these are also rings for which the identification is a truthful one. It’s probably obvious at this point from the wishlist of rings that the sufficient (although not necessary) condition I seek is that is an infinite integral domain. Indeed:
Theorem: Let be an infinite integral domain. Then, the map which takes a polynomial to its associated function is an embedding.
Although the proof isn’t too hard (for those acquainted with rings of fractions and the consequences of the Euclidean algorithm on the proof should follow fairly quickly) we shall postpone it for now.
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