Noetherian Modules: ACC, Finite Ascending Chain Definition - Bland

[Math Amateur]

I am reading Paul E. Bland's book, "Rings and Their Modules".

I am trying to understand Chapter 4, Section 4.2 on Noetherian and Artinian modules and need help with the definition of a noetherian module - in particular I need help with the nature of an ascending chain of submodule ...

Definition 4.2.1 reads as follows:

View attachment 3668

As I mentioned above, my question relates to the nature of an ascending chain - in particular, how do we regard a finite ascending chain ... indeed, is there such a thing as a finite ascending chain of submodules? (Note that the definition reads as if every ascending chain is unending or infinite?)For example, if we have a chain consisting of two submodules:

$$M_1 \subseteq M_2 $$ (1)

How to we regard this? Is it a terminating ascending chain?

Do we in fact regard (1) as an infinite ascending chain, as shown:

$$M_1 \subseteq M_2 \subseteq M_3 \subseteq M_4 \subseteq \ ... \ ... \ ...$$

where $$M_2 = M_3 = M_4 = \ ... \ ... \ ... $$So, the (essentially finite) ascending chain terminates at $$M_2$$ ... ...

Can someone clarify the above? Is this the right way to think about a finite ascending chain of submodules?

Hope someone can help ... ...

Peter

 

[Euge]

Hi Peter,

You can certainly speak of finite chains, and we have actually discussed this in the subject of modules of finite length. In any case, if in $(1)$, $M_1$ and $M_2$ are submodules of a module $M$, then the chain $M_1 \subset M_2$ can be viewed as the infinite ascending chain $M_1 \subset M_2 \subset M \subset M \subset \cdot$. You cannot always "reverse" this process, i.e., an infinite ascending chain may not be considered a finite chain. However, if you can always reverse this process in a module $M$, then $M$ is Noetherian.

 

[Math Amateur]

Hi Peter,

You can certainly speak of finite chains, and we have actually discussed this in the subject of modules of finite length. In any case, if in $(1)$, $M_1$ and $M_2$ are submodules of a module $M$, then the chain $M_1 \subset M_2$ can be viewed as the infinite ascending chain $M_1 \subset M_2 \subset M \subset M \subset \cdot$. You cannot always "reverse" this process, i.e., an infinite ascending chain may not be considered a finite chain. However, if you can always reverse this process in a module $M$, then $M$ is Noetherian.

Most helpful, Euge .. ...

Thank you,

Peter