Sum of an Indexed Family of Submodules - Northcott, pages 8-9

[Math Amateur]

I am reading D. G. Northcott's book, Lessons on Rings, Modules and Multiplicities.

On pages 8 and 9, Northcott defines/describes the sum of an indexed family of submodules, as follows:https://www.physicsforums.com/attachments/3507
View attachment 3508At the conclusion of the above text on the construction of the sum of an indexed family of submodules, Northcott writes the following:
" ... ... The submodule \(\displaystyle L\) which has just been constructed, is called the sum of the \(\displaystyle L_i \) and is denoted \(\displaystyle \sum_{i \in I} L_i\). Not only does the sum contain each of the summands \(\displaystyle L_i\), but it is clearly the smallest submodule of \(\displaystyle N\) which has this property. ... ... "
Now it is not immediately obvious to me why the sum constructed as above, should be, as Northcott claims, 'clearly' the smallest submodule that contains each of the summands.

Can someone please show me (formally and rigorously) exactly why the sum of an indexed family of modules, constructed as above, should be (clearly) the smallest submodule containing each of the summands?

Help will be appreciated ... ...

Peter

 

[Euge]

I am reading D. G. Northcott's book, Lessons on Rings, Modules and Multiplicities.

On pages 8 and 9, Northcott defines/describes the sum of an indexed family of submodules, as follows:https://www.physicsforums.com/attachments/3507
View attachment 3508At the conclusion of the above text on the construction of the sum of an indexed family of submodules, Northcott writes the following:
" ... ... The submodule \(\displaystyle L\) which has just been constructed, is called the sum of the \(\displaystyle L_i \) and is denoted \(\displaystyle \sum_{i \in I} L_i\). Not only does the sum contain each of the summands \(\displaystyle L_i\), but it is clearly the smallest submodule of \(\displaystyle N\) which has this property. ... ... "
Now it is not immediately obvious to me why the sum constructed as above, should be, as Northcott claims, 'clearly' the smallest submodule that contains each of the summands.

Can someone please show me (formally and rigorously) exactly why the sum of an indexed family of modules, constructed as above, should be (clearly) the smallest submodule containing each of the summands?

Help will be appreciated ... ...

Peter

Let $K$ be a submodule of $N$ such that $K \supseteq L_i$ for all $i\in I$. The goal is to show that $L \subseteq K$. Then since $K$ was arbitrary and $L$ is itself a submodule of $N$ containing each $L_i$, $L$ must be the smallest submodule of $N$ relative to the property of containing all the $L_i$.

Let $x \in L$. Then $x = \sum_{i\in I} x_i$, where $x_i \in L_i$ and $x_i = 0$ for all but finitely many $i$. Let $J = \{i\in I : x_i \neq 0\}$. Then $J$ is a finite set and $x = \sum_{j\in J} x_j$. Since $x_j \in L_j \subseteq K$ for all $j\in J$ and closure under addition holds in $K$, $x \in K$. Hence, $L \subseteq K$.