Proving A + B is smallest and A ∩ B is largest submodule

[Math Amateur]

I am reading T. S. Blyth's book "Module Theory: An Approach to Linear Algebra" ... ... and am currently focussed on Chapter 2: Submodules; Intersections and Sums ... ...

I need help with understanding two claims that Blyth makes concerning submodules ...

The relevant text is as follows: ( see end of post for other text that may be relevant)

I have two questions concerning the above text ... ...
Question 1In the above text we read:

" ... ... We know that $A + B$ is the smallest submodule of $M$ that contains both $A$ and $B$, ... ... "My question is: how exactly do we know this ... ? How would we formally and rigorously prove this ... ?
Question 2In the above text we read:

" ... ... and that $A \cap B$ is the largest submodule contained in both $A$ and $B$, ... ... "My question is: how exactly do we know this ... ? How would we formally and rigorously prove this ... ?Hope that someone can help with the above two questions ...

Peter

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PS Just in case readers need to reference some of Blyth's definitions or theorems in Chapter 2, I am providing the relevant text as follows:

 

[fresh_42]

" ... ... We know that $A+B$ is the smallest submodule of $M$ that contains both $A$ and $B$, ... ... "
My question is: how exactly do we know this ... ? How would we formally and rigorously prove this ... ?

How are $A+B$ and $LC(A\cup B)$ related?
Then given any submodule $C \subseteq M$ which contains both, $A$ and $B$, does it contain $LC(A\cup B)$? And why?

" ... ... and that $A \cap B$ is the largest submodule contained in both $A$ and $B$, ... ... "
My question is: how exactly do we know this ... ? How would we formally and rigorously prove this ... ?

By theorem 2.1 $A\cap B$ is a submodule of $M$. Then how are $A\cap B$ and $LC(A\cap B)$ related?
Now given any submodule $C$, that is contained in $A$ as well as in $B$, is it contained in $A\cap B$ or $LC(A\cap B)$? And why?

And at last: If you know all this about any possible submodule $C \subseteq M$, what does it tell you?

 

[andrewkirk]

Proof of Q1.
Define $A+B\equiv \{a+b\ :\ a\in A\wedge b\in B\}$. This is easily shown to be closed under addition and scalar multiplication, and hence a module. It must be a subset of $M$ because $M$ is closed under addition and scalar multiplication and every element of $A+B$ can be expressed as the addition or scalar, multiplication of elements of $A$ and/or $B$, which are elements of $M$.

Let $G_A$ and $G_B$ be generating sets for $A$ and $B$ respectively. Then, by the definition of $A+B$, $G_A\cup G_B$ is a generating set for $A+B$.

Let $C$ be a submodule of $M$ that contains $A$ and $B$. Then $C$ contains both $G_A$ and $G_B$ and hence contains $G_A\cup G_B$. Any element of $A+B$ is a finite linear combination of elements of $G_A\cup G_B$ and, since that union is in $C$ which, being a module, is closed under finite linear combinations, the element must also be in $C$. So we have $A+B\subseteq C$.

I imagine the proof for Q2 may be similar but using intersections of generating sets rather than unions. Why not have a go at that and let me know if you get stuck.

Andrew