# Which topological spaces occur as the spectrum of some ring?

• May 26th 2011, 12:53 PM
Bruno J.
Which topological spaces occur as the spectrum of some ring?
Is there a complete characterization of such spaces?
Thanks!
• May 26th 2011, 11:35 PM
Drexel28
Quote:

Originally Posted by Bruno J.
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?
• May 27th 2011, 10:32 PM
Bruno J.
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.
• July 5th 2011, 07:52 PM
NonCommAlg
Re: Which topological spaces occur as the spectrum of some ring?
Quote:

Originally Posted by Bruno J.
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"!