Why we cannot in general define the product of two submodules

Question

why we can define product of two ideals of a commutative ring,but we can't in general define the product of two submodules?

                                          ——Introduction to Commutative algebra

Answer 1

If $\mathfrak{a}$ and $\mathfrak b$ are ideals in a ring $R$, with elements $x$ and $y$ respectively, the product $xy$ makes sense: Multiplication is certainly defined for ring elements. Then (finite) sums of the form $\sum_i x_i y_i$ certainly make sense because $x_i y_i \in R$, and we can add ring elements.

This is not true, however, for modules: There is no natural way to form products of elements in modules. Remember that an $R$-module $M$ is an abelian group together with a certain action of $R$ on $M$; so given elements $m, n \in M$ and $r \in R$, the formulas $m + n$ and $rm$ make sense, while $mn$ does not.

So in short, the difference is that rings have a concept of multiplication, while modules have the concept of a ring action on module elements. There's not the same symmetry here.