I have that $M$ is an $R$-module where $R$ is commutative and unitary ring. Supposing that $J$ is an ideal of $R$, when is the set $J \cdot M$ an $R$-submodule of $M$?
I have to check the two axioms of an $R$-submodule. First that $(J\cdot M,+)$ is a subgroup of $(M,+)$ and this is the hard part. Second that for $r \in R$ and $x \in J \cdot M$, $rx \in J \cdot M$ and this is always true. But for the first axiom I don't know what to do.
$J \cdot M$ is the set $\{jm, j \in J, m \in M\}$ (as seems from where I am reading)
Thanks for help.
Based on your comment describing the context of the situation, (thank you for that, by the way) it's clear that the text intends to use the standard definition of the product:
$JM:=\{\sum j_i m_i\mid j\in J, m\in M, i\in I \text{ for a finite index set $I$ }\}$
This is what is intended when looking at any sort of ideal- or module-wise product in ring theory, precisely because the group-theoretic version ($HK:=\{hk\mid h\in H, k\in K\}$) does not produce an acceptable result in the presence of both $+$ and $\cdot$ operations.