Finitely Cogenerated Modules - Bland Definition 4.1.3, Page 105

[Math Amateur]

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 \(\displaystyle \Delta = \{ 1, 2, 3, 4 \}\) ... ... and we assume

\(\displaystyle \bigcap_\Delta M_\alpha = M_1 \cap M_2 \cap M_3 \cap M_4 = 0 \)

then of course any subset of \(\displaystyle \{ M_\alpha \}_\Delta = 0\) ... ... 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 \(\displaystyle \bigcap_\Delta M_\alpha = 0\) ... with the situation \(\displaystyle \bigcap_\Delta M_\alpha = \phi\)

 

[Math Amateur]

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:I 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 \(\displaystyle \Delta = \{ 1, 2, 3, 4 \}\) ... ... and we assume

\(\displaystyle \bigcap_\Delta M_\alpha = M_1 \cap M_2 \cap M_3 \cap M_4 = 0 \)

then of course any subset of \(\displaystyle \{ M_\alpha \}_\Delta = 0\) ... ... 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 \(\displaystyle \bigcap_\Delta M_\alpha = 0\) ... with the situation \(\displaystyle \bigcap_\Delta M_\alpha = \phi\)

I have been reflecting on Bland Definition 4.1.3 ... ... ... and I now believe I was wrong when I wrote:" ... ... To explain ... if we take \(\displaystyle \Delta = \{ 1, 2, 3, 4 \}\) ... ... and we assume

\(\displaystyle \bigcap_\Delta M_\alpha = M_1 \cap M_2 \cap M_3 \cap M_4 = 0 \)

then of course any subset of \(\displaystyle \{ M_\alpha \}_\Delta = 0\) ... ... I think ... ? ... ... "
Indeed consider the following diagram with three submodules and \(\displaystyle M_1 \cap M_2 \cap M_3 = 0\) ... ... where it is clear that \(\displaystyle M_1 \cap M_2 \neq 0\) and \(\displaystyle M_1 \cap M_3 \neq 0\) ... but \(\displaystyle M_1 \cap M_2 \cap M_3 = 0\) ... ...
View attachment 5837BUT ... I still cannot see how you can possibly get \(\displaystyle M_1 \cap M_2 \cap M_3 = 0\) with no subset satisfying \(\displaystyle M_i \cap M_j = 0\) ...

In other words it appears that to get \(\displaystyle \bigcap_\Delta M_\alpha = 0\) there must always be a finite subset \(\displaystyle F \subseteq \Delta\) such that \(\displaystyle \bigcap_F M_\alpha = 0\) ... ...

... ... so ... seems to be a meaningless concept ... so ... what is wrong with my thinking ...

... hmmm ... is it that there may be an \(\displaystyle F\) but it may not be finite ... but ... think I am grasping at straws ...Can someone please clarify the notions of cogeneration and finite cogeneration ... ...PeterPS I am still puzzled by the name co-generation for this concept ... ... which implies to me that there is something in it related to generation of modules ... but what exactly in the definition of finitely cogenerated has to do with the generation of modules ... ?

 

[steenis]

Peter, I am still in Chapter 3 of Bland's book.

About the etymology of co. It is something like this:
If we have a Property in category $C$ then the coProperty in $C$ is the dual Property in opposite category $C^o$

See definitions 4.1.2 and 4.1.5

So in category $C = \mbox{Hom} _R$ we have
$M$ is generated by $\{M_\alpha\}_\Delta$ if there is an epimorphism $\bigoplus _\Delta M_\alpha \longrightarrow M$

then in the opposite category $C^o$ we have
$M$ is generated by $\{M_\alpha\}_\Delta$ if there is a co-epimorphism $\mbox{co}(\bigoplus _\Delta M_\alpha) \mbox{co}(\longrightarrow) M$

i.e., in $C^o$ we have
$M$ is generated by $\{M_\alpha\}_\Delta$ if there is a monomorphism $\Pi _\Delta M_\alpha \longleftarrow M$

and finally in category $C = \mbox{Hom} _R$ we have
$M$ is co-generated by $\{M_\alpha\}_\Delta$ if there is a monomorphism $M\longrightarrow \Pi _\Delta M_\alpha$