References about transfinite socle series

Question

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.

Answer 1

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.