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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.

Justin Campbell
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1 Answers1

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You can do this for an arbitrary ring (with or without unit). Jacobson's original article can be found here (JSTOR, needs a university subscription).

I cannot do better than to simply quote C. Chevalley's Math Review (MathSciNet, needs a university subscription):

A (two-sided) ideal $\frak J$ in a ring $\frak A$ is called primitive if $0$ is the only $x\in\frak A$ such that ${\frak A}x\subset\frak J$. Let $S$ be the set of primitive ideals in $\frak A$; a topology is introduced in $S$ by defining the closure of a subset $M$ of $S$ to be the set of primitive ideals which contain the intersection of all ideals in $M$. The space defined in this way is a $T_0$-space, but generally not $T_1$ (except when $\frak A$ is commutative). If $\frak A$ has a unit element, $S$ is compact (that is, bicompact). The space $S$ is not changed if $\frak A$ is replaced by $\frak A/\frak R$, where $\frak R$ is the radical of $\frak A$. If $\frak A$ is semisimple and has a unit element, a necessary and sufficient condition for $S$ to be disconnected is that there should exist a decomposition of $\frak A$ as the direct sum of two two-sided ideals not equal to $\{0\}$. It is shown that, given any totally disconnected compact space $S$, it is possible to construct a ring $\frak A$ whose space of primitive ideals is homeomorphic with $S$: the elements of $S$ are certain mappings of $S$ into a field (not necessarily commutative) which can be selected arbitrarily.

t.b.
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