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Let $(A,\mathfrak{m})$ be a local noetherian domain. Suppose that $M$ and $N$ are free modules over $A$ and $f: M \rightarrow N$ is an $A$-module morphism. Suppose as well that $f$ is injective and that the induced map $\bar{f}: M/\mathfrak{m} \rightarrow N/\mathfrak{m}$ is injective as well. Then, is it true that for any $a \in A$ we have $M/a \rightarrow N/a$ is injective?

Notice that in the case that $M$ and $N$ are finitely generated (which I did not assume) this follows from Nakayama's lemma. In fact, my hypothesis on the map $\bar{f}$ says that $Q:= N/M$ has the properties that (1) it is finitely generated and (2) $Tor_1^A(Q,A/\mathfrak{m}) = (0)$. It follows that $Q$ is actually free over $A$ and so the the sequence $0 \rightarrow M \rightarrow N \rightarrow Q \rightarrow 0$ will split.

I seem to just be completely lost on the type of example one could have in the case that the modules themselves are not finitely generated. (An example with $M$ and $N$ not f.g but $Q$ is doesn't count either).

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