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)https://www.physicsforums.com/attachments/5891I have two questions concerning the above text ... ...
Question 1In the above text we read:

" ... ... We know that \(\displaystyle A + B\) is the smallest submodule of \(\displaystyle M\) that contains both \(\displaystyle A\) and \(\displaystyle B\), ... ... "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 \(\displaystyle A \cap B\) is the largest submodule contained in both \(\displaystyle A\) and \(\displaystyle B\), ... ... "
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 ...

PeterPS 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:
https://www.physicsforums.com/attachments/5892
View attachment 5893
View attachment 5894
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View attachment 5896

 

[GJA]

Hi Peter,

Question 1

In the above text we read:

" ... ... We know that \(\displaystyle A + B\) is the smallest submodule of \(\displaystyle M\) that contains both \(\displaystyle A\) and \(\displaystyle B\), ... ... "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 \(\displaystyle A \cap B\) is the largest submodule contained in both \(\displaystyle A\) and \(\displaystyle B\), ... ... "
My question is: how exactly do we know this ... ? How would we formally and rigorously prove this ... ?

For Question 1, let $K$ be any submodule of $M$ that contains both $A$ and $B$. Then $A\cup B\subseteq K$, so that $A+B=\langle A\cup B\rangle\subseteq \langle K\rangle =K$. Since $K$ was arbitrary, the statement follows.

For Question 2, let $K$ be any submodule contained in both $A$ and $B$. Then $K\subseteq A\cap B$ and it's known from a theorem in the text you posted that $A\cap B$ is a submodule contained in $A$ and $B.$ Since $K$ was arbitrary, the statement follows.

 

[Math Amateur]

Hi Peter,
For Question 1, let $K$ be any submodule of $M$ that contains both $A$ and $B$. Then $A\cup B\subseteq K$, so that $A+B=\langle A\cup B\rangle\subseteq \langle K\rangle =K$. Since $K$ was arbitrary, the statement follows.

For Question 2, let $K$ be any submodule contained in both $A$ and $B$. Then $K\subseteq A\cap B$ and it's known from a theorem in the text you posted that $A\cap B$ is a submodule contained in $A$ and $B.$ Since $K$ was arbitrary, the statement follows.

Thanks GJA ... appreciate your help

Peter

 

[steenis]

Question 1: $A+B$ is the smallest submoduie of $M$ that contains both $A$ and $B$

Of course $A\subset A+B$ and $B\subset A+B$.
Suppose $X$ is a submodule of $M$ that contains both $A$ and $B$: $A\subset X$ and $B\subset X$.
Then $A+B\subset X$.
Thus, $X$ is larger (or equal) than $A+B$.

Question 2: $A\cap B$ is the largest submodule of $M$ contained in both $A$ and $B$

Of course $A\cap B\subset A$ and $A\cap B\subset B$
Suppose $X$ is a submodule of $M$ contained in both $A$ and $B$: $X\subset A$ and $X\subset B$.
Then $X \subset A\cap B$.
Thus, $X$ is smaller (or equal) than $A\cap B$.