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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 $$M$$ is indecomposable, then we are done ... "
Is Bland arguing that if $$M$$ is indecomposable then we can regard $$M$$ 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 $$M$$ is not indecomposable, we may write $$M = X \bigoplus Y$$. At least one of $$X$$ and $$Y$$ cannot be a finite direct sum of its indecomposable submodules. ... ... "
Can someone please explain why at least one of $$X$$ and $$Y$$ cannot be a finite direct sum of its indecomposable submodules ... ... ?
... indeed ... Bland is arguing the $$M$$ is not indecomposable ... so $$M$$ is decomposable ... so $$M = X \bigoplus Y$$ ... but how does $$M$$ being decomposable stop $$X$$ and $$Y$$ 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 $$M$$ fails to have a decomposition of the form ...
$$M = M_1 \bigoplus M_2 \bigoplus \ ... \ ... \ \bigoplus M_n $$ ... ... ... ... ... (1)
... so if both of $$X$$ and $$Y$$ were finite direct sums of indecomposable submodules then $$M$$ would have a decomposition of the form (1) ... which violates the assumption that $$M$$ 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
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 $$M$$ is indecomposable, then we are done ... "
Is Bland arguing that if $$M$$ is indecomposable then we can regard $$M$$ 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 $$M$$ is not indecomposable, we may write $$M = X \bigoplus Y$$. At least one of $$X$$ and $$Y$$ cannot be a finite direct sum of its indecomposable submodules. ... ... "
Can someone please explain why at least one of $$X$$ and $$Y$$ cannot be a finite direct sum of its indecomposable submodules ... ... ?
... indeed ... Bland is arguing the $$M$$ is not indecomposable ... so $$M$$ is decomposable ... so $$M = X \bigoplus Y$$ ... but how does $$M$$ being decomposable stop $$X$$ and $$Y$$ 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 $$M$$ fails to have a decomposition of the form ...
$$M = M_1 \bigoplus M_2 \bigoplus \ ... \ ... \ \bigoplus M_n $$ ... ... ... ... ... (1)
... so if both of $$X$$ and $$Y$$ were finite direct sums of indecomposable submodules then $$M$$ would have a decomposition of the form (1) ... which violates the assumption that $$M$$ 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
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