# Finite Sum of Indecomposable Modules .... Bland, Proposition 4.2.10 .... ....

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In summary, a finite sum of indecomposable modules refers to the sum of a finite number of modules that cannot be written as a direct sum of two non-zero submodules. Bland, Proposition 4.2.10 states that any finite sum of indecomposable modules can be written as a direct sum of indecomposable modules, and this theorem is closely related to the concept of a finite sum of indecomposable modules. An indecomposable module cannot be written as a finite sum of indecomposable modules, and Bland, Proposition 4.2.10 has significance in the study of modules as it allows for the decomposition of complex structures into simpler components and has practical applications in algebraic geometry, representation theory, and
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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 4.2.10 ... ...

Proposition 4.2.10 reads as follows:View attachment 8213My questions are as follows:Question 1

In the above proof by Bland we read the following:

" ... ... If $$\displaystyle M$$ is indecomposable, then we are done ... "

Is Bland arguing that if $$\displaystyle M$$ is indecomposable then we can regard $$\displaystyle M$$ itself as a "finite sum" of indecomposable R-modules ... ... can someone please confirm that this is the case ...

Question 2

In the above proof by Bland we read the following:

" ... ... Since $$\displaystyle M$$ is not indecomposable, we may write $$\displaystyle M = X \bigoplus Y$$. At least one of $$\displaystyle X$$ and $$\displaystyle Y$$ cannot be a finite direct sum of its indecomposable submodules. ... ... "

Can someone please explain why at least one of $$\displaystyle X$$ and $$\displaystyle Y$$ cannot be a finite direct sum of its indecomposable submodules ... ... ?

... indeed ... Bland is arguing the $$\displaystyle M$$ is not indecomposable ... so $$\displaystyle M$$ is decomposable ... so $$\displaystyle M = X \bigoplus Y$$ ... but how does $$\displaystyle M$$ being decomposable stop $$\displaystyle X$$ and $$\displaystyle Y$$ both being decomposable ... ?--------------------------------------------------------------------------------------------------------------------------------------------

***EDIT***

Regarding Question 2 ... I think I should have read the proof more carefully ... and noted that Bland is assuming not only that M is not indecomposable ... but also that $$\displaystyle M$$ fails to have a decomposition of the form ...

$$\displaystyle M = M_1 \bigoplus M_2 \bigoplus \ ... \ ... \ \bigoplus M_n$$ ... ... ... ... ... (1)

... so if both of $$\displaystyle X$$ and $$\displaystyle Y$$ were finite direct sums of indecomposable submodules then $$\displaystyle M$$ would have a decomposition of the form (1) ... which violates the assumption that $$\displaystyle M$$ fails to have a decomposition of the form ...

Is that correct ...?

----------------------------------------------------------------------------------------------------------------------------------------------Help will be appreciated ...

Peter=========================================================================Definition 4.2.9 is relevant to the above post so I am providing the text of Definition 4.2.9 ... as follows ...

View attachment 8214Hope that helps ...

Peter

Last edited:
$M$ can be decomposable or indecomposable.

If $M$ is indecomposable then we are ready, because then $M$ is a finite sum of $1$ indecomposable submodule of $M$, namely $M$ itself. That answers question 1.

If $M$ is decomposable, then we have to prove that $M$ is a finite direct sum of indecomposable submodules of $M$.

For the proof we suppose that there is no finite direct sum of indecomposable submodules of $M$, i.e., $M$ cannot be written as a finite direct sum of indecomposable submodules of $M$. This will turn out to be a contradiction.

See my answer in the other post.

## 1. What is the definition of a finite sum of indecomposable modules?

A finite sum of indecomposable modules is a mathematical concept that refers to the sum of a finite number of indecomposable modules, which are modules that cannot be written as a direct sum of two non-zero submodules.

## 2. How is the finite sum of indecomposable modules related to Bland, Proposition 4.2.10?

Bland, Proposition 4.2.10 is a mathematical theorem that states that any finite sum of indecomposable modules can be written as a direct sum of indecomposable modules, with each indecomposable module appearing only once in the direct sum. This theorem is closely related to the concept of a finite sum of indecomposable modules.

## 3. Can an indecomposable module be written as a finite sum of indecomposable modules?

No, by definition, an indecomposable module cannot be written as a direct sum of two non-zero submodules. Therefore, an indecomposable module cannot be written as a finite sum of indecomposable modules.

## 4. What is the significance of Bland, Proposition 4.2.10 in the study of modules?

Bland, Proposition 4.2.10 is an important result in the study of modules as it provides a way to break down a finite sum of modules into simpler components, which can aid in understanding the structure and properties of the original module. It also allows for the classification of modules by their indecomposable components.

## 5. How can Bland, Proposition 4.2.10 be applied in real-world scenarios?

Bland, Proposition 4.2.10 has many practical applications, particularly in the field of algebraic geometry and representation theory. It can be used to decompose complex structures into simpler ones, making them easier to analyze and manipulate. It also has applications in coding theory, where modules are used to encode and decode information.

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