The state of the art on topological rings - the Jacobson topology

Question

I was recently studying the Jacobson density theorem and I found it quite interesting. Most textbooks I've seen, including Jacobson's own Basic Algebra, only spend a few lines about the reason why it is called "density theorem" and its relationship with topology, and only focus on the algebraic side.

The Wikipedia page I've linked has a short section about the topological characterization of this "density" property. Namely, if we give the set of linear endomorphism of some vector space a proper tailor-made topology, then the fact that a subring is dense, in the sense of the Jacobson density theorem, is equivalent to being a dense subset in this topology. For this reason, from now on, I shall refer to this topology on $\mathrm{End}(V)$ as the Jacobson topology.

Now, I have noticed that this topology makes $\mathrm{End}(V)$ a topological ring and a topological vector space (when such a vector space structure exists) and I sense it has some other very nice properties (which I have no intention of talking about here). I guess one could also frame the Jacobson Topology in a categorical way. Is the Jacobson topology well understood or not? Am I one of the first to investigate it close-up?

Jacobson himself doesn't seem to be spending too much time on his own topology; the Wikipedia article references in that section I've talked about the textbook Noncommutative Rings, by Herstein (1995), which is actually quite dated; nonetheless, Herstein doesn't spend more than ten lines on the topic, and just touches it as a fun observation.

I tried to dig deeper; I thought the best way to frame the topic topologically was to think of topological rings. Topological groups was probably too vague and Topological Vector Spaces was probably misleading, due to its association with Functional Analysis, rather than Algebra. The only textbook I could find entirely devoted to topological rings was Seth Warner's Topological Rings (1993). This textbook dedicates a whole chapter about the Jacobson density theorem, while at the same time, he acknowledges the fact that the whole chapter will almost entirely be algebraic in nature, and not topological. In fact, he barely even mentions topological rings in the chapter.

I couldn't find anything in the literature; or maybe I just don't know what to search.

So I wonder, has this topic already been studied? Are there any textbooks or papers who covered it in great depth and I've missed them?

If not, do you think this could be a valuable research project? Sidenote: I'm still a graduate student, and I haven't even started my PhD yet. I have already done some original research in the past, but since I'm still quite young, even a small topic such as this one could be valuable to me. Even just a small paper on the arXiv would be more than enough.

Thank you in advance, everyone.

Answer 1

This is a pretty well-established and widely studied construction in the literature on topological algebra. However, in my experience it's not always easy to navigate that literature, which explains how you ended up here!

This is an instance of what is called the finite topology on the endomorphism ring of a module. You'll probably have a lot more luck if you search that term. If you're interested in seeing some collected basic facts about this in the special case of endomorphisms of vector spaces, you might take a look at section 2 of the following:

Iovanov, Miodrag C.; Mesyan, Zachary; Reyes, Manuel L., Infinite-dimensional diagonalization and semisimplicity, Isr. J. Math. 215, No. 2, 801-855 (2016). ZBL1372.16031. (also at arXiv)

Among the references there, some textbooks that can be helpful when first learning about this are the following:

Dăscălescu, Sorin; Năstăsescu, Constantin; Raianu, Şerban, Hopf algebras. An introduction, Pure and Applied Mathematics, Marcel Dekker. 235. New York, NY: Marcel Dekker. ix, 401 p. (2001). ZBL0962.16026.

Warner, Seth, Topological rings, North-Holland Mathematics Studies. 178. Amsterdam: North-Holland. x, 498 p. (1993). ZBL0785.13008.