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 ... "

(R is a ring with 1, M is a left module over R, A is a subset of M)

Can anyone formally and explicitly prove this statement?

Peter