MHB Jordan-Holder Theorem for Modules .... .... Another Two Questions ....

  • Thread starter Thread starter Math Amateur
  • Start date Start date
  • Tags Tags
    Modules Theorem
Math Amateur
Gold Member
MHB
Messages
3,920
Reaction score
48
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
====================================================================================
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:
Physics news on Phys.org
(4.3) and (4.4) are compostion series of $M$, by definition $M_i$ is a maximal submodules of $M_{i-1}$, that is $ M_{i-1}/M_i$ is simple.
Therefore $M/M_1$ and $M/N_1$ are simple because $M = M_0 = N_0$.

In (4.7), since $M/M_1$ is simple, the isomorphic $N_1/(M_1 \cap N_1)$ must be simple, therefore $(M_1 \cap N_1)$ is maximal in $N_1$, idem $(M_1 \cap N_1)$ is maximal in $M_1$, using (4.8).

$(M_1 \cap N_1)$ is a submodule of $M$, and prop.4.2.5 says that submodules of noetherian modules are noetherian, idem arterian.
 
I asked online questions about Proposition 2.1.1: The answer I got is the following: I have some questions about the answer I got. When the person answering says: ##1.## Is the map ##\mathfrak{q}\mapsto \mathfrak{q} A _\mathfrak{p}## from ##A\setminus \mathfrak{p}\to A_\mathfrak{p}##? But I don't understand what the author meant for the rest of the sentence in mathematical notation: ##2.## In the next statement where the author says: How is ##A\to...
The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
When decomposing a representation ##\rho## of a finite group ##G## into irreducible representations, we can find the number of times the representation contains a particular irrep ##\rho_0## through the character inner product $$ \langle \chi, \chi_0\rangle = \frac{1}{|G|} \sum_{g\in G} \chi(g) \chi_0(g)^*$$ where ##\chi## and ##\chi_0## are the characters of ##\rho## and ##\rho_0##, respectively. Since all group elements in the same conjugacy class have the same characters, this may be...
Back
Top