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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) ... ...

Proposition 4.2.16 reads as follows:

View attachment 8243

https://www.physicsforums.com/attachments/8244

"... ... 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 \(\displaystyle N_1\) ... ... ? ... ... )

"... ... 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

Hope access to the above helps ...

Peter

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) ... ...

Proposition 4.2.16 reads as follows:

View attachment 8243

https://www.physicsforums.com/attachments/8244

*Near the middle of the above proof (top of page 116) we read the following:***Question 1**"... ... 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 \(\displaystyle N_1\) ... ... ? ... ... )

*Near the middle of the above proof (top of page 116) we read the following:***Question 2**"... ... 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

====================================================================================

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

Hope access to the above helps ...

Peter

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