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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
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
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