Suppose that $I$ is a two-sided ideal in the ring $R$, and that $M$ is a module over the quotient ring $R/I$. Why can we naturally regard $M$ as a $R$-module that is annihilated by $I$? Conversely, suppose that $M$ is a $R$-module annihilated by $I$. Why is $M$ naturally a module over $A/I$?
The obvious module action is $rm:=(r+ I)m$.
Conversely, given an ideal $I$ and an $R$ module $M$ we would like to use $(r+ I)m:= rm$, but this is not well-defined unless $I$ is contained in the annihilator of $M$.