Just reflecting on my own question - it is easy to show that RA is a submodule of M, but what was worrying me was 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
We need to show that
If then for since N is a submodule
and then the addition of these elements, visually, also is in N (if two elements belong to a submodule then so does the element that is formed by their addition)
So ... ... (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?