Let $I$ be an ideal in a ring $B$ with $I^2=0$. Furthermore one knows that one has a splitting
$\alpha: B/I \rightarrow B$
of the natural projection.
Let $M$ be a finitely generated module over $B/I$.
Set $N:= M \otimes_{B/I} I$, where we let $I$ be a module over $B/I$ by setting
$\bar{b}i:=\alpha(\bar{b})i$ for $\bar{b}\in B/I$ and $i\in I $.
Question: is every $B-$module homomorphism $M \rightarrow N$ injective?
Here I take $M$ to be a $B-$module via $B\rightarrow B/I$.