Is there a complete characterization of such spaces?

Thanks!

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- May 26th 2011, 11:53 AMBruno J.Which topological spaces occur as the spectrum of some ring?
Is there a complete characterization of such spaces?

Thanks! - May 26th 2011, 10:35 PMDrexel28
- May 27th 2011, 09:32 PMBruno 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.

- Jul 5th 2011, 06:52 PMNonCommAlgRe: Which topological spaces occur as the spectrum of some ring?
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 $\displaystyle X$ is spectral if and only if $\displaystyle X= \varprojlim Y_n,$ where $\displaystyle Y_n$ are finite and $\displaystyle T_0.$

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"!