Loewy series of modules

Question

I am studying the loewy series of module. But I can't understand how the series building.

I know that the socle is the sum of all simple submodules, but I can't understand the limit ordinal case.

Answer 1

I know that the socle is the sum of all simple submodules, but I can't understand the limit ordinal case.

If you understand the part of the definition that says $M_{\alpha+1}/M_\alpha=\mathrm{Soc}(M/M_\alpha)$, then the thing to understand is that using this to inductively define the series, one successor at a time, cannot exceed the first limit ordinal above $\alpha$ (call it $\beta$.) You will get closer and closer to $\beta$, but not reach it.

So, there has to be a method for "leaping the gap" after defining all those things before $\beta$, and the way to do it is to take the union of all the $M_\lambda$ for $\lambda < \beta$. The result is $M_\beta$, which you've now defined for that limit ordinal.

Then continues the process with $M_{\beta+1}, M_{\beta+2},M_{\beta+3}\ldots$ and so on, but you cannot reach the next limit ordinal (call it $\gamma$) this way. You have to take the union over all $\lambda < \gamma$ of $M_\gamma$, and then you've got $M_\gamma$.

This process is called transfinite induction and it allows you to define $M_\alpha$ for every ordinal $\alpha$.

If you still have problems understanding transfinite induction, I invite you to read the wiki and or seek basic problems and answers about the topic.