Jordan-Holder Theorem for Modules .... ....

In summary, the conversation discusses Proposition 4.2.16 from Paul E. Bland's book "Rings and Their Modules," specifically the proof of part of the proposition. The question is about the statement that if $M_1 \neq N_1$, then $M_1 + N_1 = M$ because $N_1$ is a maximal submodule of $M$. The explanation given is that since $M_1$ is not equal to $N_1$, their sum is a submodule of $M$ containing the maximal submodule $N_1$ and must therefore be equal to $M$.
  • #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 help to fully understand the proof of part of Proposition 4.2.16 (Jordan-Holder) ... ...

Proposition 4.2.16 reads as follows:
View attachment 8240
View attachment 8241

Near the middle of the above proof (top of page 116) we read the following:

"... ... If \(\displaystyle M_1 \neq N_1\) then \(\displaystyle M_1 + N_1 = M\) since \(\displaystyle N_1\) is a maximal submodule of \(\displaystyle M\). ... ... "

Can someone please explain exactly how \(\displaystyle N_1\) being a maximal submodule of \(\displaystyle M\) implies that \(\displaystyle M_1 + N_1 = M\) ... ... ?

Peter
 
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  • #2
Hi Peter,

Peter said:
Near the middle of the above proof (top of page 116) we read the following:

"... ... If \(\displaystyle M_1 \neq N_1\) then \(\displaystyle M_1 + N_1 = M\) since \(\displaystyle N_1\) is a maximal submodule of \(\displaystyle M\). ... ... "

Since $M_{1}\neq N_{1}$, $M_{1}+N_{1}$ is a submodule of $M$ containing the maximal submodule $N_{1}$ and so must be $M$.
 
  • #3
GJA said:
Hi Peter,
Since $M_{1}\neq N_{1}$, $M_{1}+N_{1}$ is a submodule of $M$ containing the maximal submodule $N_{1}$ and so must be $M$.
Thanks for the help, GJA ...

Peter
 

FAQ: Jordan-Holder Theorem for Modules .... ....

What is the Jordan-Holder Theorem for Modules?

The Jordan-Holder Theorem for Modules is a fundamental result in abstract algebra that describes the structure of a module in terms of its submodules. It states that every module has a unique composition series, and any two composition series for the same module have the same length and the same composition factors, up to permutation and isomorphism.

What is a composition series for a module?

A composition series for a module is a sequence of submodules that cannot be further divided into smaller submodules. In other words, it is a chain of submodules where each subsequent submodule is a proper submodule of the previous one, and the final submodule in the chain is the trivial submodule (containing only the zero element).

How does the Jordan-Holder Theorem relate to the Jordan-Holder Theorem for Groups?

The Jordan-Holder Theorem for Modules is an extension of the Jordan-Holder Theorem for Groups, which states that every group has a unique composition series. The difference is that the Jordan-Holder Theorem for Modules deals with modules, which are more general algebraic structures than groups.

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

The Jordan-Holder Theorem for Modules is an important result in abstract algebra, as it allows us to classify modules based on their composition factors. This is useful in understanding the structure of modules and their relationship to other algebraic structures. It also has applications in other areas of mathematics, such as representation theory and homological algebra.

Is the Jordan-Holder Theorem for Modules applicable to all modules?

Yes, the Jordan-Holder Theorem for Modules is applicable to all modules. However, it is often used in the context of finite-dimensional modules over a commutative ring. In this case, the composition series can be thought of as a generalization of the concept of a prime factorization for integers.

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