I've encountered a few possible definitions of a "connected ring" and am having some confusion relating them. The first one is defined for any commutative ring:
A commutative connected ring has a spectrum which is connected in the Zariski topology.
But there is also the concept of a topological ring, where the ring itself (not the spectrum) is endowed with a topology. In this case, it also seems natural to consider a topological ring to be "connected" if the the ring itself is a connected topological space (irrespective of the spectrum). My concrete questions are thus:
- For a commutative topological ring, is there a relationship between "connectedness" in the spectrum-sense vs in the ring topology-sense? More broadly, is there any relationship between the topologies on the two spaces? They seem unrelated to me.
- When I read results on ring theory, I'm often confused whether topological statements refer to the ring itself or the spectrum. Take for example the following:
Every compact Hausdorff ring is totally disconnected. I'm assuming here that the ring in question is a topological ring, and "compact Hausdorff"-ness is refering to the topological properties of the ring itself. But here, is "total disconnected"-ness referring to the ring topology, or to the spectrum topology that results in the definition of a "connected ring" above?
Am I getting tripped up on overloaded terminology here, or is there some deeper connection that I'm missing? Thanks for the help!
Best Answer
There is not really any significant relationship between these two notions. The contexts in which you care about them tend to be very different from each other such that it is almost always obvious which one is being talked about in context. For one thing, if your ring does not have a topology, then only the first meaning can be intended. On the flip side, if someone is talking about a topological ring (and mentioning the topology), they almost certainly mean connectedness of that topology and if they wanted to talk about the other notion they would instead talk about the spectrum of the ring being connected to avoid confusion. (This includes the specific example you are asking about: "every compact Hausdorff ring is totally disconnected" is talking about the topology of the ring.) Generally, connectedness in the ring-theoretic sense is something you would care about when doing algebraic geometry (and your ring is probably closely related to rings of polynomials) and connectedness of a topological ring is something you would care about when studying the general theory of topological rings.