Math Amateur
Gold Member
MHB
- 3,920
- 48
I am reading Dummit and Foote's book: "Abstract Algebra" (Third Edition) ...
I am currently studying Chapter 10: Introduction to Module Theory ... ...
I need some help with an aspect of Dummit and Foote's Section 10.3 Basic Generation of Modules, Direct Sums and Free Modules ... ...
The start of Section 10.3 reads as follows:View attachment 8224
View attachment 8225
In the above text from Dummit and Foote we read the following:
" ... ... for submodules $$N_1, \ ... \ ... \ , N_n$$ of $$M$$, $$N_1 + \ ... \ ... \ + N_n$$ is just the submodule generated by the set $$N_1 \cup \ ... \ ... \ \cup N_n$$ ... ... "I tried to prove the above ... but despite the relationship seeming intuitive, I found it difficult to formulate a rigorous proof ...
My attempt at a proof is as follows ...
Firstly we note that $$RN$$ where $$N = \cup_{ s = 1 }^n N_s $$ is as follows:$$RN = \{ r_1 n_1 + r_2 n_2 + \ ... \ ... \ + r_p n_p \mid r_s \in R, n_s \in N , p \in \mathbb{N} \}$$
To show $$\sum_{ s = 1 }^n N_s = RN$$ ... ... ... ... ... (1)
Firstly show that $$\sum_{ s = 1 }^n N_s \subseteq RN$$ ... ... ... ... ... (i)
Now ... $$x \in \sum_{ s = 1 }^n N_s \Longrightarrow x = x_1 + x_2 + \ ... \ ... \ + x_n$$ where $$x_s \in N_s$$But ... each $$x_s \in N_s \subseteq M \Longrightarrow x_s = r'_s m'_s$$$$\Longrightarrow x = r'_1 m'_1 + r'_2 m'_2 + \ ... \ ... \ + r'_n m'_n$$ ... where $$r'_s \in R$$ and $$m'_s \in N_s$$ ...$$\Longrightarrow x \in RN$$ ...Is that correct?
Now ... to show that $$RN \subseteq \sum_{ s = 1 }^n N_s$$$$x \in RN$$ $$\Longrightarrow x = r_1 n_1 + r_2 n_2 + \ ... \ ... \ + r_p n_p $$ where $$r_s \in R$$, $$n_s \in N = \cup_{ s = 1 }^n N_s \text{ and } p \in \mathbb{N} $$... BUT! ... the $$n_s$$ do not necessarily belong to $$N_s$$ ... ?How do we proceed ...?Can someone please help ...Help will be much appreciated ...
Peter
I am currently studying Chapter 10: Introduction to Module Theory ... ...
I need some help with an aspect of Dummit and Foote's Section 10.3 Basic Generation of Modules, Direct Sums and Free Modules ... ...
The start of Section 10.3 reads as follows:View attachment 8224
View attachment 8225
In the above text from Dummit and Foote we read the following:
" ... ... for submodules $$N_1, \ ... \ ... \ , N_n$$ of $$M$$, $$N_1 + \ ... \ ... \ + N_n$$ is just the submodule generated by the set $$N_1 \cup \ ... \ ... \ \cup N_n$$ ... ... "I tried to prove the above ... but despite the relationship seeming intuitive, I found it difficult to formulate a rigorous proof ...
My attempt at a proof is as follows ...
Firstly we note that $$RN$$ where $$N = \cup_{ s = 1 }^n N_s $$ is as follows:$$RN = \{ r_1 n_1 + r_2 n_2 + \ ... \ ... \ + r_p n_p \mid r_s \in R, n_s \in N , p \in \mathbb{N} \}$$
To show $$\sum_{ s = 1 }^n N_s = RN$$ ... ... ... ... ... (1)
Firstly show that $$\sum_{ s = 1 }^n N_s \subseteq RN$$ ... ... ... ... ... (i)
Now ... $$x \in \sum_{ s = 1 }^n N_s \Longrightarrow x = x_1 + x_2 + \ ... \ ... \ + x_n$$ where $$x_s \in N_s$$But ... each $$x_s \in N_s \subseteq M \Longrightarrow x_s = r'_s m'_s$$$$\Longrightarrow x = r'_1 m'_1 + r'_2 m'_2 + \ ... \ ... \ + r'_n m'_n$$ ... where $$r'_s \in R$$ and $$m'_s \in N_s$$ ...$$\Longrightarrow x \in RN$$ ...Is that correct?
Now ... to show that $$RN \subseteq \sum_{ s = 1 }^n N_s$$$$x \in RN$$ $$\Longrightarrow x = r_1 n_1 + r_2 n_2 + \ ... \ ... \ + r_p n_p $$ where $$r_s \in R$$, $$n_s \in N = \cup_{ s = 1 }^n N_s \text{ and } p \in \mathbb{N} $$... BUT! ... the $$n_s$$ do not necessarily belong to $$N_s$$ ... ?How do we proceed ...?Can someone please help ...Help will be much appreciated ...
Peter
Last edited: