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Math Help - When is a ring of functions an integral domain?

  1. #1
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    When is a ring of functions an integral domain?

    If you have a ring of functions from an open, connected subset of a metric space to another metric space, what is the most relaxed condition you can place on the functions that will make that ring an integral domain? Is it that the functions have to be analytic?
    I only said metric space in order to exclude corner topological cases, but if you could relax that condition too, how would you do it?
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    Senior Member Tinyboss's Avatar
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    What ring operations do you want to use? If it's pointwise addition and multiplication, then you want a ring structure on the range--a metric space doesn't necessarily support that. If the range has no zero divisors, then what you need to worry about is two nonzero functions whose zero-sets unioned together are the whole domain.
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    As I suspected, I hadn't put in too much thought into phrasing my question. I'm sorry about that. I do want the operations to be pointwise addition and multiplication, and it made sense in my head because I'm really thinking about functions whose range is a subset of the reals, which is also a field apart from being a metric space. I then threw in the generalization (metric space) as an afterthought because I'm trying get a better understanding of the general case. I'm not trying to work out any particular problem here.

    You touched on what I want to discuss though: what kind of restrictions must you make on the functions to prevent zero divisors (let's restrict ourselves to f: R -> R)? It's obvious to me that continuous and even infinitely differentiable functions don't seem to work. Is the next step to require analytic functions?
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    Senior Member Tinyboss's Avatar
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    Real analytic would do it, since their zeros don't accumulate. But there are other ways. Consider the ring of functions generated by f_n(x) for n\in\mathbb{Z}, whose value is 0 for x in [n,n+1] and 1 elsewhere.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by eeyore View Post
    As I suspected, I hadn't put in too much thought into phrasing my question. I'm sorry about that. I do want the operations to be pointwise addition and multiplication, and it made sense in my head because I'm really thinking about functions whose range is a subset of the reals, which is also a field apart from being a metric space. I then threw in the generalization (metric space) as an afterthought because I'm trying get a better understanding of the general case. I'm not trying to work out any particular problem here.

    You touched on what I want to discuss though: what kind of restrictions must you make on the functions to prevent zero divisors (let's restrict ourselves to f: R -> R)? It's obvious to me that continuous and even infinitely differentiable functions don't seem to work. Is the next step to require analytic functions?
    You'll be fine as long as you pick classes whose kernels are nowhere dens--for example real analytic will do.


    Edit: Dang it, Tinyboss!!
    Last edited by Chris L T521; April 26th 2011 at 11:00 PM.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by eeyore View Post
    If you have a ring of functions from an open, connected subset of a metric space to another metric space, what is the most relaxed condition you can place on the functions that will make that ring an integral domain? Is it that the functions have to be analytic?
    I only said metric space in order to exclude corner topological cases, but if you could relax that condition too, how would you do it?
    I don't know how much you are into this topic, but I can't ever help promoting my favorite books. A very, very, very good book (albeit a little difficult) is the book Rings of Continuous Functions-Gillman and Jerrison. It could answer any conceivable algebraic/analytic/topological/set theoretic question you have regarding rings of continuous functions. It's probably in your library.
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    Quote Originally Posted by Drexel28 View Post
    I don't know how much you are into this topic, but I can't ever help promoting my favorite books. A very, very, very good book (albeit a little difficult) is the book Rings of Continuous Functions-Gillman and Jerrison. It could answer any conceivable algebraic/analytic/topological/set theoretic question you have regarding rings of continuous functions. It's probably in your library.
    Thanks. I'm pretty interested in the topic and I'll check it out even though we're using Conway's book on complex analysis from this series, and I hate it very much. =)
    Last edited by eeyore; April 27th 2011 at 05:34 AM. Reason: Added quote
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