I'm trying to investigate the notion of primitive spectrum and its so-called Jacobson or hull-kernel topology, but I can only find references which define it for C*-algebras: see the Wikipedia page "Spectrum of a C*-algebra" for the definition I'm talking about. It seems like this definition would make sense for any (let's stick with unital) ring whatsoever, so I suspect the problem is that in full generality we don't actually get a topology.
So here's what I want to ask: can you help me think of an example of a unital ring for which the Jacobson topology on its primitive spectrum is not actually a topology? Or even better, does anyone know of general conditions under which the primitive spectrum has a natural topology? Also, when the Jacobson topology is defined, is the primitive spectrum always quasi-compact?
By way of motivation, certain complex algebras have come up in my research in representation theory. Explaining what they are would take me far afield, but they do have the following nice property. In their paper "Extensions of representations of $p$-adic nilpotent groups," S. Gelfand and D. Kazhdan call a complex unital algebra $A$ quasi-finite provided that it has a filtration $A_0 \subset A_1 \subset \cdots \subset A$ by finite-dimensional semisimple subalgebras $A_k$, and simple modules for each $A_k$ are finite-dimensional.