Finitely Generated Modules and Maximal Submodules

  • I
  • Thread starter Math Amateur
  • Start date
  • Tags
    Modules
In summary, the conversation discusses Proposition 6.1.2 and its proof, which relies on Zorn's Lemma and the notion of inductive sets. The question is asked about why Bland needs to show that ##\bigcup_\mathscr{C}## is a proper submodule of ##M## that contains ##N##. It is clarified that in the case of an infinite chain, there may not be a largest submodule, so ##\bigcup_\mathscr{C}## serves as an upper bound. The expert confirms that this understanding is correct.
  • #1
Math Amateur
Gold Member
MHB
3,998
48
I am reading Paul E. Bland's book, "Rings and Their Modules".

I am focused on Section 6.1 The Jacobson Radical ... ...

I need help with an aspect of Proposition 6.1.2 ... ...

Proposition 6.1.2 relies on Zorn's Lemma and the notion of inductive sets ... ... so I am providing a short note from Bland on Zorn's Lemma and inductive sets ... ... as follows:
?temp_hash=9ae88151a81f41f5b8cb4e744b816927.png

NOTE: My apologies for the poor quality of the above image - due to some over-enthusiastic highlighting of Bland's text
frown.png

Now, Proposition 6.1.2 reads as follows:

?temp_hash=9ae88151a81f41f5b8cb4e744b816927.png

Now ... in the above proof of Proposition 6.1.2, Bland writes the following:"... ... If ##\mathscr{C}## is a chain of submodules of ##\mathscr{S}##, then ##x_1 \notin \bigcup_\mathscr{C}## , so ##\bigcup_\mathscr{C}## is a proper submodule of ##M## and contains ##N##. Hence ##\mathscr{S}## is inductive ... ...My question is as follows: Why does Bland bother to show that ## \bigcup_\mathscr{C}## is a proper submodule of ##M## that contains ##N## ... presumably he is showing that any chain of submodules in ##\mathscr{S}## has an upper bound ... is that right?
... ... but why does he need to do this as the largest submodule in the chain would be an upper bound ... ... ?Hope someone can help ... ...

PeterNOTE: My apologies for not being able to exactly reproduce Bland's embellished S in the above text ...
 

Attachments

  • Bland - Zorn's Lemma and Inductive Sets.png
    Bland - Zorn's Lemma and Inductive Sets.png
    41.6 KB · Views: 853
  • Bland - Prposition 6.1.2.png
    Bland - Prposition 6.1.2.png
    41.6 KB · Views: 786
Last edited:
Physics news on Phys.org
  • #2
Math Amateur said:
... ... but why does he need to do this as the largest submodule in the chain would be an upper bound ... ... ?

Why would there be a largest one?
 
  • Like
Likes Math Amateur
  • #3
micromass said:
Why would there be a largest one?
Well I was thinking of the finite case ... e.g. where for example, \mathcal{C} might be

##N'_1 \subseteq N'_2 \subseteq N'_3##

so ... ##N'_3## in this case is an upper bound on the chain ##\mathcal{C}## ... BUT ... your question me me think that my thinking does not cover the case of an infinite chain ...

In the case of an infinite chain there may be no largest submodule and so we need to have ##\bigcup_\mathcal{C} N'## as an upper bound ...Can you confirm that my thinking is now correct ...

Peter
 
  • Like
Likes fresh_42
  • #4
Yes.

The upper bound of ##\mathscr{C}## need not be in ##\mathscr{C}##, but is has to be in ##\mathscr{S}##.
 

FAQ: Finitely Generated Modules and Maximal Submodules

What is a finitely generated module?

A finitely generated module is a module that can be generated by a finite set of elements. This means that all elements of the module can be expressed as linear combinations of the finite set of generators.

What is the difference between a finitely generated module and a free module?

A free module is a module that has a basis, meaning that its elements are linearly independent and can generate the entire module. On the other hand, a finitely generated module may not have a basis and may require more than one set of generators to span the entire module.

What is a maximal submodule?

A maximal submodule is a submodule that is not a proper submodule of any other submodule. In other words, it is the largest possible submodule that can exist within a given module.

Can a finitely generated module have more than one maximal submodule?

Yes, a finitely generated module can have multiple maximal submodules. This can occur when the module has more than one set of generators, each of which corresponds to a different maximal submodule.

How are finitely generated modules and maximal submodules related?

A maximal submodule is a special type of submodule that is not a proper submodule of any other submodule. Finitely generated modules can have multiple maximal submodules, but not all submodules of a finitely generated module are necessarily maximal submodules.

Back
Top