Sums and Intersections of Submodules .... Berrick and Keating Exercise 1.2.12

[Math Amateur]

I am reading An Introduction to Rings and Modules With K-Theory in View by A.J. Berrick and M.E. Keating (B&K).

I need help with Exercise 1.2.12 ...

Exercise 1.2.12 reads as follows:
https://www.physicsforums.com/attachments/5102
Can someone please help me to get started on this problem ...

Help will be much appreciated ...

Peter

 

[Deveno]

You have to show two things:

1) If $N'$ is a submodule of $M$ such that $M_i \subseteq N'$ for all $i \in I$, that $N \subseteq N'$.

2) If $L'$ is a submodule of $M$ such that $L' \subseteq M_i$ for all $i \in I$, that $L' \subseteq L$.

The second property is well-nigh obvious, if $L' \subseteq M_i$ for each $i$, it is surely in $\bigcap\limits_i M_i$, by the definition of intersection.

Of course, it may help to prove that $\bigcap\limits_i M_i$ is indeed a submodule of $M$.

Number 1) is a little trickier, you may wish to show instead that $N$ is the smallest submodule of $M$ containing the set:

$\bigcup\limits_i M_i$ hint: use the closure of module addition.