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. 



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> 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 $\overline{\text{soc}}(M)=\text{soc}^{\mu(M)}(M)$.)
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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.



References about transfinite socle series
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**](http://dx.doi.org/10.1016/0022-4049(92)90008-4), J. Pure Appl. Algebra 82, No. 1, 27-37 (1992). [ZBL0786.16002](https://zbmath.org/?q=an:0786.16002),


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.


