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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 $A/I$?

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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 $M$.

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