Suppose I have a commutative ring with identity $R$, and an $R$-module $M$. Next I have an $R$-submodule $N$ of $M$. Finally, I have a multiplicatively closed subset $S$ of $R$.
An element $s\in S$ can be multiplied onto an element $m\in M$ by the way $M$ was defined.
What about a coset? Is the natural way to multiply $s$ by an element $m + N$ of $M/N$ just the obvious one?
That is, $s(m + N) = (sm) + N$?