Generating/spanning modules and submodules .... .... Blyth Theorem 2.3

[Math Amateur]

I am reading T. S. Blyth's book: Module Theory: An Approach to Linear Algebra ...

I am focused on Chapter 2: Submodules; intersections and sums ... and need help with the proof of Theorem 2.3 ...

Theorem 2.3 reads as follows:View attachment 8157In the above proof we read the following:

" ... ... A linear combination of elements of \(\displaystyle \bigcup_{ i \in I }\) is precisely a sum of the form \(\displaystyle \sum_{ j \in J } m_j\) for some \(\displaystyle J \in P(I).\) ... ... "But ... Blyth defines a linear combination as in the text below ...https://www.physicsforums.com/attachments/8158So ... given the above definition wouldn't a linear combination of elements of \(\displaystyle \bigcup_{ i \in I } M_i\) be a sum of the form \(\displaystyle \sum_{ j \in J } \lambda_j m_j\) ... and not just \(\displaystyle \sum_{ j \in J } m_j\) ... ... ?
Hope someone can help ...

Peter

 

[steenis]

Hint: if $m \in M$ and $M$ is a module, then also $\lambda m \in M$, for $\lambda \in R$.

Thus $\Sigma_{j \in J} \lambda_j m'_j = \Sigma_{j \in J} m_j $ for $m_j = \lambda_j m'_j \in M_j$

However, the set $S$ used above, is not a module ...