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I am reading Paul E. Bland's book, "Rings and Their Modules".
I am trying to understand Section 4.1 on Generating and Cogenerating Classes ... ...
In particular I am trying to understand and get a good sense of the notion of a module being finitely cogenerated ...
Bland defines a finitely cogenerated module as follows:https://www.physicsforums.com/attachments/5835I seem to be misunderstanding the notion of finite cogeneration of a module ... indeed at first glance the notion seems pretty meaningless ...
To explain ... if we take $$\Delta = \{ 1, 2, 3, 4 \}$$ ... ... and we assume
$$\bigcap_\Delta M_\alpha = M_1 \cap M_2 \cap M_3 \cap M_4 = 0 $$
then of course any subset of $$\{ M_\alpha \}_\Delta = 0$$ ... ... I think ... ?... and so the concept seems empty ... ?
Can someone please explain the concept of cogeneration ... and hence finite cogeneration ...
... and how it relates to the generation of modules ...Peter
EDIT
In hindsight, I think I may be muddling the situation $$\bigcap_\Delta M_\alpha = 0$$ ... with the situation $$\bigcap_\Delta M_\alpha = \phi$$
I am trying to understand Section 4.1 on Generating and Cogenerating Classes ... ...
In particular I am trying to understand and get a good sense of the notion of a module being finitely cogenerated ...
Bland defines a finitely cogenerated module as follows:https://www.physicsforums.com/attachments/5835I seem to be misunderstanding the notion of finite cogeneration of a module ... indeed at first glance the notion seems pretty meaningless ...
To explain ... if we take $$\Delta = \{ 1, 2, 3, 4 \}$$ ... ... and we assume
$$\bigcap_\Delta M_\alpha = M_1 \cap M_2 \cap M_3 \cap M_4 = 0 $$
then of course any subset of $$\{ M_\alpha \}_\Delta = 0$$ ... ... I think ... ?... and so the concept seems empty ... ?
Can someone please explain the concept of cogeneration ... and hence finite cogeneration ...
... and how it relates to the generation of modules ...Peter
EDIT
In hindsight, I think I may be muddling the situation $$\bigcap_\Delta M_\alpha = 0$$ ... with the situation $$\bigcap_\Delta M_\alpha = \phi$$
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