Theorem 2.3: Submodule Generation by Family of Submodules - T. S. Blyth

[Math Amateur]

I am reading T. S. Blyth's book "Module Theory: An Approach to Linear Algebra" ... ... and am currently focussed on Chapter 1: Modules, Vector Spaces and Algebras ... ...

I need help with a basic and possibly simple aspect of Theorem 2.3 ...

Since the answer to my question may depend on Blyth's previous definitions and theorems I am providing some relevant text from Blyth prior to Theorem 2.3 ... but those confident with the theory obviously can go straight to the theorem at the bottom of the scanned text ...

Theorem 2.3 together with some relevant prior definitions and theorems reads as follows: (Theorem 2,3 at end of text fragment)View attachment 5886
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In the above text (near the end) we read, in the statement of Theorem 2.3:

" ... ... then the submodule generated by \(\displaystyle \bigcup_{ i \in I } M_i\) consists of all finite sums of the form \(\displaystyle \sum_{ j \in J } m_j\) ... ... "The above statement seems to assume we take one element from each \(\displaystyle M_j\) in forming the sum \(\displaystyle \sum_{ j \in J } m_j\) ... ... but how do we know a linear combination does not take more than one element from a particular \(\displaystyle M_j\) , say \(\displaystyle M_{ j_0 }\) ... ... or indeed all elements from one particular \(\displaystyle M_j\) ... rather than one element from each submodule in the family \(\displaystyle \{ M_i \}_{ i \in I }\) ...

Hope someone can clarify this ...

Peter

 

[Deveno]

To briefly address your concerns, recall that module addition is commutative and associative, so we can group all terms in our finite sum from anyone $M_i$ so they are adjacent, and since $M_i$ is closed under module addition and $R$-multiplication, we can "combine" all those terms into a single element $m_i$.

Of course we may have just a single (non-zero) "term" in the sum $\sum\limits_{j \in J} m_j$, because $J$ may be a singleton subset of $I$ (which is non-empty).

 

[Math Amateur]

To briefly address your concerns, recall that module addition is commutative and associative, so we can group all terms in our finite sum from anyone $M_i$ so they are adjacent, and since $M_i$ is closed under module addition and $R$-multiplication, we can "combine" all those terms into a single element $m_i$.

Of course we may have just a single (non-zero) "term" in the sum $\sum\limits_{j \in J} m_j$, because $J$ may be a singleton subset of $I$ (which is non-empty).

... thanks Deveno ... that clarified the matter ...

... appreciate your help ...

Peter