Results 1 to 4 of 4

Math Help - Which topological spaces occur as the spectrum of some ring?

  1. #1
    MHF Contributor Bruno J.'s Avatar
    Joined
    Jun 2009
    From
    Canada
    Posts
    1,266
    Thanks
    1
    Awards
    1

    Which topological spaces occur as the spectrum of some ring?

    Is there a complete characterization of such spaces?
    Thanks!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by Bruno J. View Post
    Is there a complete characterization of such spaces?
    Thanks!
    I'm not sure if there is a general answer but I know you can restrict it quite a bit--no? For example, aren't the only Hausdorff ones those which come from finite fields?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor Bruno J.'s Avatar
    Joined
    Jun 2009
    From
    Canada
    Posts
    1,266
    Thanks
    1
    Awards
    1
    I don't know about that specific statement (I'm not sure what you mean - the spectrum of a field is trivial...), but yeah, you can definitely narrow it down a lot. Which leads one to wonder if there is a general criterion to decide.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7

    Re: Which topological spaces occur as the spectrum of some ring?

    Quote Originally Posted by Bruno J. View Post
    Is there a complete characterization of such spaces?
    Thanks!
    yes, there are many of them. these topological spaces are called spectral spaces. a nice characterization is in terms of inverse limit of topological

    spaces: a topological space X is spectral if and only if X= \varprojlim Y_n, where Y_n are finite and T_0.

    i think now that you have a name, you can find other characterizations of spectral spaces.

    by the way, i've assumed that by "ring" you meant "commutative ring"!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Topological Spaces
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: May 19th 2010, 09:14 PM
  2. Topological Spaces
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: March 4th 2010, 10:01 AM
  3. about topological spaces
    Posted in the Differential Geometry Forum
    Replies: 6
    Last Post: June 30th 2009, 04:28 AM
  4. Hoeomorphic topological spaces
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: March 19th 2009, 04:55 PM
  5. question about topological spaces
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: May 19th 2008, 06:23 PM

Search Tags


/mathhelpforum @mathhelpforum