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

of $M$, denoted by $\text{soc}(M)$, to be the sum of all simple $R$-submodules of $M$. We create thesocle$$\text{Soc}(M):=\big(\text{soc}^\alpha(M):\alpha\text{ is an ordinal}\big)$$ 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 thetransfinite socle filtration$\text{soc}_\alpha(M)$ 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 $\overline{\text{soc}}(M)=\text{soc}^{\mu(M)}(M)$.)socle factor

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.