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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
Question 1Near the middle of the above proof (top of page 116) we read the following:
"... ... so $$M_1 \cap N_1$$ is a maximal submodule of $$M_1$$ and $$N_1$$, since $$M/M_1$$ and $$M/N_1$$ are simple modules. ... ... "Can someone please explain why $$M/M_1$$ and $$M/N_1$$ being simple modules implies that $$M_1 \cap N_1$$ is a maximal submodule of $$M_1$$ and $$N_1$$ ... ... ?( ***NOTE*** : I can see that $$M/M_1$$ and $$M/N_1$$ being simple modules implies that $$M_1$$ and $$N_1$$ are maximal ... but how does that imply that $$M_1 \cap N_1$$ is a maximal submodule of $$M_1$$ and $$N_1$$ ... ... ? ... ... )
Question 2Near the middle of the above proof (top of page 116) we read the following:
"... ... Using Proposition 4.2.14 we see that $$M $$ is artinian and noetherian and Proposition 4.2.5 indicates that $$M_1 \cap N_1$$ is artinian and noetherian ... ... "Can someone please explain how/why Proposition 4.2.5 indicates that $$M_1 \cap N_1$$ 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
Question 1Near the middle of the above proof (top of page 116) we read the following:
"... ... so $$M_1 \cap N_1$$ is a maximal submodule of $$M_1$$ and $$N_1$$, since $$M/M_1$$ and $$M/N_1$$ are simple modules. ... ... "Can someone please explain why $$M/M_1$$ and $$M/N_1$$ being simple modules implies that $$M_1 \cap N_1$$ is a maximal submodule of $$M_1$$ and $$N_1$$ ... ... ?( ***NOTE*** : I can see that $$M/M_1$$ and $$M/N_1$$ being simple modules implies that $$M_1$$ and $$N_1$$ are maximal ... but how does that imply that $$M_1 \cap N_1$$ is a maximal submodule of $$M_1$$ and $$N_1$$ ... ... ? ... ... )
Question 2Near the middle of the above proof (top of page 116) we read the following:
"... ... Using Proposition 4.2.14 we see that $$M $$ is artinian and noetherian and Proposition 4.2.5 indicates that $$M_1 \cap N_1$$ is artinian and noetherian ... ... "Can someone please explain how/why Proposition 4.2.5 indicates that $$M_1 \cap N_1$$ 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
Last edited: