MHB Noetherian Modules .... Cohn Theorem 2.2 .... ....

  • 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 P.M. Cohn's book: Introduction to Ring Theory (Springer Undergraduate Mathematics Series) ... ...

I am currently focused on Section 2.2: Chain Conditions ... which deals with Artinian and Noetherian rings and modules ... ...

I need help with understanding an aspect of the proof of Theorem 2.2 ... ...Theorem 2.2 and its proof (including some preliminary relevant definitions) read as follows:
View attachment 8003
https://www.physicsforums.com/attachments/8004

At the end of the above proof by Cohn we read the following:

" ... ... If $$a_j \in N_{i_j} $$ and $$k = \text{ max} \{ i_1, \ ... \ ... \ , i_r \}$$, then equality holds in our chain from N_k onwards. ... ... "
Can someone please explain how/why $$a_j \in N_{i_j} $$ and $$k = \text{ max} \{ i_1, \ ... \ ... \ , i_r \}$$ implies that equality holds in our chain from $$N_k$$ onwards. ... ... ?Help will be appreciated ...

Peter
 
Last edited:
Physics news on Phys.org
Hi Peter,

Each $a_j$ is an element of $N=\bigcup N_i$, and therefore an element of some $N_i$, say $N_{i_j}$. If $k = \max \{ i_1, \ldots, i_r \}$, then $N_k$ contains all the $a_j$; as these elements generate $N$ and $N_k\subset N$, we have $N_k=N$.

Now, for any $i\ge k$, we have $N_k=N \subset N_i\subset N$, and therefore $N_i=N_k=N$.
 
castor28 said:
Hi Peter,

Each $a_j$ is an element of $N=\bigcup N_i$, and therefore an element of some $N_i$, say $N_{i_j}$. If $k = \max \{ i_1, \ldots, i_r \}$, then $N_k$ contains all the $a_j$; as these elements generate $N$ and $N_k\subset N$, we have $N_k=N$.

Now, for any $i\ge k$, we have $N_k=N \subset N_i\subset N$, and therefore $N_i=N_k=N$.
Thanks castor28 ...

Think I follow that argument ...

Just reflecting further to make sure I fully understand ...

Thanks again,

Peter
 
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