Submodules A + B and A intersect B .... Blyth Ch. 2

[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

==============================================================================

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