Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Show a set of functions such that f(x)=0 is maximal ideal of the set of all real func

  1. Jun 6, 2012 #1
    Given:
    Denote a ring G = { all functions from ℝ --> ℝ }
    And set N = { all functions from ℝ --> ℝ such that for any x in ℝ, f(x) = 0 }

    Want to prove:
    To prove that N is the maximal ideal of G by showing that the quotient ring G / N is isomorphic to the set of real numbers ℝ

    ** What I'm still getting stuck at this point is the following:

    1/ What should an element of G / N look like. Since G contains all real function, is it true that an element in G / N should be something like f(x) + N, where f(x) is in ℝ ? or is it something else ?

    2/ My approach is to try to define a map, say f, from G/N to ℝ. But then I get stuck on what (in ℝ) should I send elements of G/N to ?

    For first approach, I try to send everything in G/N to f(0). But then I recognize that by the mean of "all functions", G may have some functions which are not defined at 0, say f(x) = 1/x, so f(0) doesn't work.

    For second try, I try to send everything in G/N to f(1). But again, the same issue as my first approach comes up again.


    Would someone please help me on this problem?
    Thanks in advance. o:)
     
  2. jcsd
  3. Jun 6, 2012 #2
    Re: Show a set of functions such that f(x)=0 is maximal ideal of the set of all real

    Your description of N isn't right. You're saying that a function f is in N if for any x, f(x) = 0. So N consists of the zero function and nothing more. That's not what's intended here. You're misinterpreting what the problem says.

    Your first step would be to re-read the problem and clarify what N is supposed to be.


    Want to prove:
    You mean "a" maximal ideal. I'm being picky because these kinds of exercises are designed to teach you to think precisely about the concepts. A ring may have many maximal ideals.


    In other words, by invoking the fact that a ring modulo an ideal is a field iff the ideal is maximal. Is that part already clear to you?

    Another important bit of pickiness. f is a function. f(x) is the value of the function at the point x. We'll sometimes casually ignore the difference, and there's no harm in it. But this is one of those times when the distinction is important. f(x) is a real number, not a function. A coset would look like f + N.


    What you need to do is first clarify the definition of N; then go through the exercise of proving that it is indeed an ideal. Once you do that, the rest of this will become more clear.

    Also, by the way, is G supposed to be all functions? Or possibly all continuous functions? I've usually seen the result you're trying to prove stated with continuous functions; and I don't happen to know if it works out for arbitrary functions.
     
    Last edited: Jun 6, 2012
  4. Jun 6, 2012 #3

    mathwonk

    User Avatar
    Science Advisor
    Homework Helper

    Re: Show a set of functions such that f(x)=0 is maximal ideal of the set of all real

    you are right that this problem is easy if stated correctly, and just as easy for all functions as for continuous ones. the less easy direction is the converse, that all maximal ideals have this form, and that uses continuity and compactness.
     
  5. Jun 6, 2012 #4
    Re: Show a set of functions such that f(x)=0 is maximal ideal of the set of all real

    Thanks for your responses.
    Yes, I'm familiar with this idea. I think the goal of this problem is to show this, since ℝ is a field.

    ** By how the person, who put up this question, restates the question for me, I have some updates :

    1/ Ring G indeed contains all real functions. There is no notion of continuous, invertible ...
    2/ I have an interpretation that set N is not the set of zero functions. It contains functions f(x) such that when I take a number x in ℝ, then I get 0 when evaluate the function at x.

    So I think an example is if I take x = 3, then the function x - 3 is in N, because when I evaluate it at 3, the result is 0. But x - 3 is not the zero function.

    Using f(x) = x + 3 as an example, is it true that when I mod this element with N, to get an element in G/ N, it will look something like (x + 3) + N ? But then if I want to send this element to ℝ, what should the output be ? I'm thinking about sending to f(x), but ....

    I'm a bit confused :confused::confused:
     
  6. Jun 6, 2012 #5

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Re: Show a set of functions such that f(x)=0 is maximal ideal of the set of all real

    This is still very awkwardly stated. But I think you mean either that N is "the set of all functions such that f(x)= 0 for some x", not for "all x", or that N is "the set of all functions such that f(x0)= 0 for some specific x0".

     
    Last edited: Jun 6, 2012
  7. Jun 6, 2012 #6
    Re: Show a set of functions such that f(x)=0 is maximal ideal of the set of all real

    To me it looks like you need to clarify whether or not the definition of N depends on some pre-chosen point x_0. If so, then your notation should reflect the difference between the "constant" x_0 and the "variable" x.
     
  8. Jun 6, 2012 #7

    mathwonk

    User Avatar
    Science Advisor
    Homework Helper

    Re: Show a set of functions such that f(x)=0 is maximal ideal of the set of all real

    try letting N be the set of functions such that f(0) = 0.
     
  9. Jun 6, 2012 #8
    Re: Show a set of functions such that f(x)=0 is maximal ideal of the set of all real



    Either you're having a hard time trying to understand your own question's terms (and the material and etc.), or else you're having

    some difficulties with the language.

    I guess that, just as in stachexchange, if you post your question in your own language it may be a chance somebody will translate

    it and make it clear to everybody.

    Finally, if you meant [tex]N_x:=\{f:\mathbb{R}\to\mathbb{R}\,\,|\,\,\text{for some fixed}\,x\in\mathbb{R}\,,\,f(x)=0\}[/tex] then you only need to check the quotient ring [itex]G/N_x[/itex] is a field (and a rather common one)...

    DonAntonio
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Show a set of functions such that f(x)=0 is maximal ideal of the set of all real func
  1. Maximal ideals in Z[X] (Replies: 1)

Loading...