# Jordan-Holder Theorem for Modules .... .... Another Two Questions ....

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In summary, Propositions 4.2.14 and 4.2.5 state that $M/M_1$ and $M/N_1$ are simple modules, implying that $M_1 \cap N_1$ is maximal in both $M_1$ and $N_1$. Additionally, Proposition 4.2.5 states that submodules of noetherian and artinian modules are also noetherian and artinian, respectively. Therefore, $(M_1 \cap N_1)$ is a maximal submodule of both $M_1$ and $N_1$.
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I am reading Paul E. Bland's book, "Rings and Their Modules".

I am focused on Section 4.2: Noetherian and Artinian Modules and need some further help to fully understand the proof of part of Proposition 4.2.16 (Jordan-Holder) ... ...

View attachment 8243
https://www.physicsforums.com/attachments/8244
Question 1Near the middle of the above proof (top of page 116) we read the following:

"... ... so $$\displaystyle M_1 \cap N_1$$ is a maximal submodule of $$\displaystyle M_1$$ and $$\displaystyle N_1$$, since $$\displaystyle M/M_1$$ and $$\displaystyle M/N_1$$ are simple modules. ... ... "Can someone please explain why $$\displaystyle M/M_1$$ and $$\displaystyle M/N_1$$ being simple modules implies that $$\displaystyle M_1 \cap N_1$$ is a maximal submodule of $$\displaystyle M_1$$ and $$\displaystyle N_1$$ ... ... ?( ***NOTE*** : I can see that $$\displaystyle M/M_1$$ and $$\displaystyle M/N_1$$ being simple modules implies that $$\displaystyle M_1$$ and $$\displaystyle N_1$$ are maximal ... but how does that imply that $$\displaystyle M_1 \cap N_1$$ is a maximal submodule of $$\displaystyle M_1$$ and $$\displaystyle N_1$$ ... ... ? ... ... )
Question 2Near the middle of the above proof (top of page 116) we read the following:

"... ... Using Proposition 4.2.14 we see that $$\displaystyle M$$ is artinian and noetherian and Proposition 4.2.5 indicates that $$\displaystyle M_1 \cap N_1$$ is artinian and noetherian ... ... "Can someone please explain how/why Proposition 4.2.5 indicates that $$\displaystyle M_1 \cap N_1$$ is artinian and noetherian ... ...
Help will be appreciated ...

Peter
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The above post refers to Propositions 4.2.14 and 4.2.5 ... so I am providing text of the statements of the propositions as follows:View attachment 8245https://www.physicsforums.com/attachments/8246

Peter

Last edited:
(4.3) and (4.4) are compostion series of $M$, by definition $M_i$ is a maximal submodules of $M_{i-1}$, that is $M_{i-1}/M_i$ is simple.
Therefore $M/M_1$ and $M/N_1$ are simple because $M = M_0 = N_0$.

In (4.7), since $M/M_1$ is simple, the isomorphic $N_1/(M_1 \cap N_1)$ must be simple, therefore $(M_1 \cap N_1)$ is maximal in $N_1$, idem $(M_1 \cap N_1)$ is maximal in $M_1$, using (4.8).

$(M_1 \cap N_1)$ is a submodule of $M$, and prop.4.2.5 says that submodules of noetherian modules are noetherian, idem arterian.

## 1. What is the Jordan-Holder Theorem for Modules?

The Jordan-Holder Theorem for Modules states that any two composition series for a finite length module are equivalent, meaning that they have the same length and the same composition factors up to isomorphism.

## 2. What is the significance of the Jordan-Holder Theorem for Modules?

The Jordan-Holder Theorem for Modules is significant because it allows us to classify modules up to isomorphism, making it a useful tool in the study of abstract algebra and representation theory.

## 3. How is the Jordan-Holder Theorem for Modules related to the Jordan-Holder Theorem for Groups?

The Jordan-Holder Theorem for Modules is a generalization of the Jordan-Holder Theorem for Groups, which states that any two composition series for a finite group are equivalent. The module version extends this result to modules, which are more general algebraic structures than groups.

## 4. Can the Jordan-Holder Theorem for Modules be applied to infinite length modules?

No, the Jordan-Holder Theorem for Modules only applies to finite length modules. Infinite length modules have more complex composition series and cannot be classified in the same way as finite length modules.

## 5. What are some applications of the Jordan-Holder Theorem for Modules?

The Jordan-Holder Theorem for Modules has many applications, including in the study of representation theory, group theory, and algebraic geometry. It is also used in the classification of simple modules and in proving the existence of maximal submodules.

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