Filtering a module by square zero extension

Question

It is claimed: Let $A^\eta \rightarrow A$ be a square zero extension of commutative ring. i.e. kernel $I^2=0$. Let $X$ be $A^\eta$-module. Then we can obtain a filtration $$ 0 \rightarrow X' \rightarrow X \rightarrow X'' \rightarrow 0$$

of $A^\eta$ modules such that $I$ acts trivially on $X', X''$.

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Some comments would be appreciated.

Answer 1

Just let $X'=IX$ and $X''=X/IX$. Then $IX''=0$ trivially, and $IX'=I^2X=0$ since $I^2=0$.