Is RA the Smallest Submodule of M Containing A?

[Math Amateur]

On page 351 Dummit and Foote make the following statement:

"It is easy to see using the Submodules' Criterion that for any subset A of M, RA is indeed a submodule of M and is the smallest submodule of M which contains A ... "

I am not sure how to (formally and explicitly) prove this statement.

However, reflecting on the above, it is easy to show that RA is a submodule of M, but what is worrying me is the formal proof that RA is the smallest submodule of M that contains A.

However following the statement:

"It is easy to see using the Submodules' Criterion that for any subset A of M, RA is indeed a submodule of M and is the smallest submodule of M which contains A ... " Dummit and Foote write:

"i.e. any submodule of M which contains A also contains RA"

[I still find it perplexing that this actually shows that RA is the smallest submodule of M which contains A but anyway ... this is, I think, not hard to prove ...}

So to show that any submodule of M which contains A also contains RA

Let N be a submodule of M such that $$A \subseteq N$$

We need to show that $$RA \subseteq N$$

Let $$x \in RA$$

Now $$RA = \{ r_1a_1 + r_2a_2 + ... ... + r_ma_m \ | \ r_1, r_2, ... ... , r_m \in R, \ a_1, a_2, ... ... , a_m \in A, m \in \mathbb{Z}^{+}$$

So $$x = r_1a_1 + r_2a_2 + ... ... + r_ma_m$$ for $$r_1, r_2, ... ... , r_m \in R, \ a_1, a_2, ... ... , a_m \in A$$

If $$A \subseteq N$$ then $$r_ia_i \in N$$ for $$1 \le i \le n$$ since N is a submodule

and then the addition of these elements, visually, $$r_1a_1 + r_2a_2 + ... ... + r_ma_m$$ also is in N (if two elements belong to a submodule then so does the element that is formed by their addition)

So $$x \in N$$

Thus $$x \in RA \Longrightarrow x \in N$$

So $$A \subseteq N \Longrightarrow RA \subseteq N$$ ... ... (1)

However, I am still not completely sure how to formally show that RA is the smallest submodule of M that contains A i.e. how does the implication (1) demonstrate this - can someone help by showing this explicitly and formally?

Peter

[This has also been posted on MHF]

 

[Fernando Revilla]

However, I am still not completely sure how to formally show that RA is the smallest submodule of M that contains A i.e. how does the implication (1) demonstrate this - can someone help by showing this explicitly and formally?

Consider the set
$$\mathcal{N}=\{N\subseteq M:N\text{ submodule of }M\text{ and }A\subseteq N \}$$
Then, $\subseteq$ is an order relation on $\mathcal{N}$, $RA\in \mathcal{N}$ and $RA\subseteq N$ for all $N\in\mathcal{N}.$ This means that $RA$ is the smallest element related to $(\mathcal{N},\subseteq)$

 

[Deveno]

One can argue by contradiction, here.

Suppose we have a submodule \(\displaystyle N\) with:

\(\displaystyle A \subseteq N \subsetneq RA\).

By your previous argument, since \(\displaystyle N\) is a submodule containing \(\displaystyle A\), we have:

\(\displaystyle RA \subseteq N \subsetneq RA\) that is:

\(\displaystyle RA \neq RA\), a contradiction. So no such \(\displaystyle N\) can exist.