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$\DeclareMathOperator\FWM{FWM}\DeclareMathOperator\End{End}$For any ring $R$ there's an algebra of finite width matrices with entries in $R$. By finite width matrices I mean the ones that have only finitely many nonzero rows. Let's call this algebra $\FWM(R)$. One may add the unit matrix to it to make it unital. $\FWM(R)$ is naturally a subalgebra of $\End \bigoplus_{\Bbb N} R$. $\End \bigoplus_{\Bbb N} R$ is also known as column-finite matrices, because it can be realised as the algebra of matrices that have finitely many nonzero entries in each column.

While $\End \bigoplus_{\Bbb N} R$ is pretty well-known and studied, $\FWM(R)$ appears to be quite mysterious; I haven't found any texts concerning its structure and/or categories of modules of any flavour. So, I'd like to know whether it was already studied (...it definitely was, but I didn't manage to find a way to figure out how people named this algebra) and what is already known about it.

Here are several questions that interest me particularly:

  1. Does $\FWM(R)$ have any estabilished name?

  2. What are the properties of left and right natural actions of $\FWM(R)$ on $\bigoplus_{\Bbb N} R$ and $\End \bigoplus_{\Bbb N} R$. Are they flat on either side?

  3. What are the ways to define $\FWM(R)$ in a "coordinate-free" way, similar to its bigger cousin $\End \bigoplus_{\Bbb N} R$? Or, slightly differently put, does it have any universal properties?

  4. Are there any nontrivial left or right simple modules over $\FWM(R)$?

I became interested in this algebra because I have a hunch that it should naturally act on the right on the space of coarse functions on any countable homogeneous coarse space. Typical example of such space is any countable group with its natural coarse structure. I had no success in proving that, and most approaches withered at the point where universal categorical reasoning met with ad hoc infinite matrix constructions. In particular, I suppose that there should be a bimodule structure on the space of functions from $\Bbb N$ to $R$ taking finitely many values (i. e. functions pulled back via a map from $\Bbb N$ to a finite set) but I currently have no understanding of the nature of this bimodule structure. In a sense, $\FWM(R)$ action should serve as an underlying structure of a coarse structure; this underlying structure being a choice of embedding of countable discrete set (countable group) into its Stone-Cech compactification, and "meaningful" part being the orbit decomposition under left regular action of group on itself.

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  • $\begingroup$ Isnt this the ideal of endomorphisms with image contained in a finitely generated submodule of the infinite direct sum. When R is a division ring these would be the finite rank endomorphisms. I guess if R is Noetherian on the correct side then this would be the endomorphisms with finitely generated image. People certainly look at such things. $\endgroup$
    – Benjamin Steinberg
    Commented Mar 26, 2023 at 23:02
  • $\begingroup$ @BenjaminSteinberg Well, yes, for matrix realization it's exactly finite rank endomorphisms; or, as one may say, algebra of compact operators for natural colimit topology. But I haven't found anything relevant! And it feels strange, because quotient by this ideal is very well studied Calkin algebra. Maybe I should have elaborated on meaning of "coordinate-free"; it would be very nice to get it as full endomorphism algebra of some natural thing (other than itself as a free rank one module). // Also I'm perfectly happy with R being a field. $\endgroup$
    – Denis T
    Commented Mar 26, 2023 at 23:23
  • $\begingroup$ If R is commutative and Noetherian then finite rank is a coordinate free description where finite rank means finitely generated image. $\endgroup$
    – Benjamin Steinberg
    Commented Mar 27, 2023 at 10:34
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    $\begingroup$ If you look up the structure of primitive rings with a minimal left ideal you will find they are precisely the suhrings of column finite matrices over a division ring containing all the finite rank matrices. This is a consequence of Jacobson's density theorem and can be found in Cohn's algebra book. $\endgroup$
    – Benjamin Steinberg
    Commented Mar 27, 2023 at 10:39
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    $\begingroup$ These days ring theorists don't tend to look at nonunital rings so one should probably look at older books $\endgroup$
    – Benjamin Steinberg
    Commented Mar 27, 2023 at 10:41

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