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I am reading Paul E. Bland's book, "Rings and Their Modules".
I am trying to understand Chapter 4, Section 4.2 on Noetherian and Artinian modules and need help with the proof of $$ (3) \Longrightarrow (1) $$ in Proposition 4.2.3.
Proposition 4.2.3 and its proof read as follows:
View attachment 3660
View attachment 3661
The first line of the proof of $$ (3) \Longrightarrow (1) $$ reads as follows:
"If $$ M_1 \subseteq M_2 \subseteq M_3 \subseteq \ ... \ $$ is an ascending chain of submodules of M then $$\bigcup_{ i = 1 }^{ \infty } M_i $$ is a finitely generated module of $$M$$. ... ... "
My question is as follows:
How do we know that if
... ... $$ M_1 \subseteq M_2 \subseteq M_3 \subseteq \ ... \ $$ is an ascending chain of submodules of $$M$$
then
... ... $$\bigcup_{ i = 1 }^{ \infty }$$ is a finitely generated module of $$M$$ ...
That is ... why exactly does this follow?
Peter
I am trying to understand Chapter 4, Section 4.2 on Noetherian and Artinian modules and need help with the proof of $$ (3) \Longrightarrow (1) $$ in Proposition 4.2.3.
Proposition 4.2.3 and its proof read as follows:
View attachment 3660
View attachment 3661
The first line of the proof of $$ (3) \Longrightarrow (1) $$ reads as follows:
"If $$ M_1 \subseteq M_2 \subseteq M_3 \subseteq \ ... \ $$ is an ascending chain of submodules of M then $$\bigcup_{ i = 1 }^{ \infty } M_i $$ is a finitely generated module of $$M$$. ... ... "
My question is as follows:
How do we know that if
... ... $$ M_1 \subseteq M_2 \subseteq M_3 \subseteq \ ... \ $$ is an ascending chain of submodules of $$M$$
then
... ... $$\bigcup_{ i = 1 }^{ \infty }$$ is a finitely generated module of $$M$$ ...
That is ... why exactly does this follow?
Peter
Last edited: