Can anyone explain to me why the minimal prime ideals of a commutative ring (with the subspace topology inherited from the Zariski topology) form a totally disconnected space, or give a reference? I remember that this is true but can't seem to prove it myself or find the proof anywhere.
I would be especially happy if there is some proof that does not use the (easy) fact that this space is Hausdorff. This is because I am trying to prove that the primitive spectrum of a certain noncommutative ring, which I know is not Hausdorff, is totally disconnected. Hopefully the proof works in my situation when phrased in the appropriate way.