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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 \(\displaystyle N_1, \ ... \ ... \ , N_n\) of \(\displaystyle M\), \(\displaystyle N_1 + \ ... \ ... \ + N_n\) is just the submodule generated by the set \(\displaystyle N_1 \cup \ ... \ ... \ \cup N_n\) ... ... "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 \(\displaystyle RN\) where \(\displaystyle N = \cup_{ s = 1 }^n N_s \) is as follows:\(\displaystyle RN = \{ r_1 n_1 + r_2 n_2 + \ ... \ ... \ + r_p n_p \mid r_s \in R, n_s \in N , p \in \mathbb{N} \}\)

To show \(\displaystyle \sum_{ s = 1 }^n N_s = RN\) ... ... ... ... ... (1)

Firstly show that \(\displaystyle \sum_{ s = 1 }^n N_s \subseteq RN\) ... ... ... ... ... (i)

Now ... \(\displaystyle x \in \sum_{ s = 1 }^n N_s \Longrightarrow x = x_1 + x_2 + \ ... \ ... \ + x_n\) where \(\displaystyle x_s \in N_s\)But ... each \(\displaystyle x_s \in N_s \subseteq M \Longrightarrow x_s = r'_s m'_s\)\(\displaystyle \Longrightarrow x = r'_1 m'_1 + r'_2 m'_2 + \ ... \ ... \ + r'_n m'_n\) ... where \(\displaystyle r'_s \in R\) and \(\displaystyle m'_s \in N_s\) ...\(\displaystyle \Longrightarrow x \in RN\) ...

Now ... to show that \(\displaystyle RN \subseteq \sum_{ s = 1 }^n N_s\)\(\displaystyle x \in RN\) \(\displaystyle \Longrightarrow x = r_1 n_1 + r_2 n_2 + \ ... \ ... \ + r_p n_p \) where \(\displaystyle r_s \in R\), \(\displaystyle n_s \in N = \cup_{ s = 1 }^n N_s \text{ and } p \in \mathbb{N} \)... BUT! ... the \(\displaystyle n_s\) do not necessarily belong to \(\displaystyle N_s\) ... ?

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 \(\displaystyle N_1, \ ... \ ... \ , N_n\) of \(\displaystyle M\), \(\displaystyle N_1 + \ ... \ ... \ + N_n\) is just the submodule generated by the set \(\displaystyle N_1 \cup \ ... \ ... \ \cup N_n\) ... ... "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 \(\displaystyle RN\) where \(\displaystyle N = \cup_{ s = 1 }^n N_s \) is as follows:\(\displaystyle RN = \{ r_1 n_1 + r_2 n_2 + \ ... \ ... \ + r_p n_p \mid r_s \in R, n_s \in N , p \in \mathbb{N} \}\)

To show \(\displaystyle \sum_{ s = 1 }^n N_s = RN\) ... ... ... ... ... (1)

Firstly show that \(\displaystyle \sum_{ s = 1 }^n N_s \subseteq RN\) ... ... ... ... ... (i)

Now ... \(\displaystyle x \in \sum_{ s = 1 }^n N_s \Longrightarrow x = x_1 + x_2 + \ ... \ ... \ + x_n\) where \(\displaystyle x_s \in N_s\)But ... each \(\displaystyle x_s \in N_s \subseteq M \Longrightarrow x_s = r'_s m'_s\)\(\displaystyle \Longrightarrow x = r'_1 m'_1 + r'_2 m'_2 + \ ... \ ... \ + r'_n m'_n\) ... where \(\displaystyle r'_s \in R\) and \(\displaystyle m'_s \in N_s\) ...\(\displaystyle \Longrightarrow x \in RN\) ...

**Is that correct?**Now ... to show that \(\displaystyle RN \subseteq \sum_{ s = 1 }^n N_s\)\(\displaystyle x \in RN\) \(\displaystyle \Longrightarrow x = r_1 n_1 + r_2 n_2 + \ ... \ ... \ + r_p n_p \) where \(\displaystyle r_s \in R\), \(\displaystyle n_s \in N = \cup_{ s = 1 }^n N_s \text{ and } p \in \mathbb{N} \)... BUT! ... the \(\displaystyle n_s\) do not necessarily belong to \(\displaystyle N_s\) ... ?

*Can someone please help ...Help will be much appreciated ...***How do we proceed ...?**Peter

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