Is a special module homomorphism injective?

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 $B\rightarrow B/I$.

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Unless $N=0$, the answer is "no" -- simply because the $0$-homomorphism will not be injective.