From Dummit and Foot, Abstract Algebra ch10.2:
Defn: Let A,B be submodules of the R-module M. The sum of A and B is the set
$A+B=\{a+b| a\in A, b\in B \}$ it is easily checked that the sum of two submodules A and B is a submodule and is the smallest submodule which contains both A and B.
By the submodule criterion, it follow that $A+B$ is a submodule of M. How can we show that that $A+B$ is the smallest submodule that contains A and B?