# References about transfinite socle series

I would like to know if there is any literature that discusses "transfinite socle series" of a ring module. Below is my attempt at defining the series.

Let $$R$$ be an associative unital ring and $$M$$ a unitary left $$R$$-module. Define the socle of $$M$$, denoted by $$\text{soc}(M)$$, to be the sum of all simple $$R$$-submodules of $$M$$. We create the transfinite socle filtration $$\text{Soc}(M):=\big(\text{soc}^\alpha(M):\alpha\text{ is an ordinal}\big)$$ of a given $$R$$-module $$M$$ as follows. First, define $$\text{soc}^0(M):=0$$ and $$\text{soc}_0(M):=0$$. If an ordinal number $$\alpha>0$$ has an immediate predecessor $$\beta$$, then $$\text{soc}_\alpha(M):=\text{soc}\big(M\big/\text{soc}^{\beta}(M)\big)\,,$$ and $$\text{soc}^\beta(M)$$ is the preimage $$\pi_\alpha^{-1}\big(\text{soc}_\alpha(M)\big)$$ of the socle factor $$\text{soc}_\alpha(M)$$ under the canonical projection $$\pi_\alpha:M\twoheadrightarrow \big(M\big/\text{soc}^{\beta}(M)\big)$$. If $$\alpha$$ is a limit ordinal, then $$\text{soc}^\alpha(M)$$ to be $$\bigcup\limits_{\beta<\alpha}\,\text{soc}^\beta(M)$$, and set $$\text{soc}_\alpha(M):=0$$. Finally, define $$\overline{\text{soc}}(M)$$ to be the union of all submodules $$\text{soc}^\alpha(M)$$ of $$M$$. (Clearly, there exists a smallest ordinal $$\mu(M)$$ such that $$\overline{\text{soc}}(M)=\text{soc}^{\mu(M)}(M)$$.)

I do not have a specific question, but I would like to learn about any information regarding transfinite socle series. For example, if $$\omega$$ is the least infinite ordinal, then does it hold that $$\text{soc}_{\omega+1}(M)=0$$ for any $$R$$-module $$M$$ (in particular, when $$R$$ is an algebra over a field $$\mathbb{K}$$, and maybe when $$M$$ is a countable-dimensional vector space over the same field $$\mathbb{K}$$)? If this is not true (for a general $$R$$, or for the case where $$R$$ is an algebra over a field), what are counterexamples? Any reference, comment, and knowledge about transfinite socle filtration will be greatly appreciated.

• I found quite a lot of literature by Googling "transfinite socle series". – Jeremy Rickard Jun 13 '20 at 11:16
• @JeremyRickard Not sure what you meant by a lot, but I only found two. (Originally, I called such a filtration simply an "infinite socle series," and Google search was not very helpful. Then, a user suggested that I changed the term to "transfinite socle series." Googling this term results in better references, but there are only two viable links.) – Batominovski Jun 13 '20 at 11:36
• I'm not sure what you count as "viable", but most of the Google results I get in (at least) the first three pages of results seem relevant. – Jeremy Rickard Jun 13 '20 at 11:43

As I said in comments, there is a fair amount of literature to be found by Googling "infinite socle series".

More specifically, a module $$M$$ for which (in the notation of the question) $$\overline{\text{soc}}(M)=M$$ is called a "semi-artinian module", and a ring $$R$$ for which every module is semi-artinian (or equivalently for which $$R$$ is semi-artinian as a module for itself) is called a "semi-artinian ring". There is quite a lot of literature to be found on semi-artinian rings and modules.

For the specific question asked at the end of the question, the following example is adapted from Section 5 of

Nguyen V. Dung; Smith, Patrick F., On semi-artinian $$V$$-modules, J. Pure Appl. Algebra 82, No. 1, 27-37 (1992). ZBL0786.16002,

although that paper is about a rather specific class of modules, and so it would not surprise me if similar examples were known previously, or if there are simpler examples.

Let $$\mathbb{K}$$ be a field, let $$R_n=\mathbb{K}[t]/(t^n)$$ for $$n\geq1$$, and let $$R$$ be the subring of $$\prod_{n\geq1}R_n$$ consisting of elements $$(r_n)_{n\geq1}$$ such that, for some $$a\in\mathbb{K}$$, $$r_n=a$$ for all but finitely many $$n$$. Then $$R$$ is a countable dimensional $$\mathbb{K}$$-algebra, with $$\bigoplus_{n\geq1}R_n$$ as a codimension one ideal.

Let $$M=R$$, the regular $$R$$-module.

For $$k\in\mathbb{N}$$, $$\text{soc}^kM=\bigoplus_{n\geq1}\text{soc}_{R_n}^kR_n,$$ and so, since $$\text{soc}_{R_n}^n R_n=R_n$$, $$\text{soc}^\omega M=\bigoplus_{n\geq1}R_n,$$ and so $$M/\text{soc}^\omega M=\text{soc}_{\omega+1}M$$ is one-dimensional.