injection modulo jacobson radical is in general not an injection before

Let \(f: M \rightarrow N\) be a module-homomorphism between finitely generated \(A\)-modules.
Let \(\mathfrak{a} \subseteq \mathrm{Jac}(A)\) be an ideal contained in the Jacobson radical such that \(f \otimes_{A} A/\mathfrak{a}\) is an injection.

Then \(f\) is in general not an injection.

Let \(A = \mathbb{Z}_{(2)}\), \(M = A\) and \(N = \mathbb{Z}_{(2)}/2 = \mathbb{Z}/2 \mathbb{Z}\).
Let \(f: M \rightarrow N\) be the unique module homomorphism which maps \(1\) to \(1\) (observe that \(M\) is a free \(A\)-module) and let \(\mathfrak{a} = \mathfrak{m}\) be contained in the jacobson radical, as \(\mathbb{Z}_{(2)}\) is a local ring.

Then \(f\) is not injective, e.g. as \(M\) contains infinitely many elements and \(N\) is finite, but \(f \otimes_{A} A/\mathfrak{m}\) is an isomorphism.