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Let $R$ be a commutative ring with identity and $M$ an $R$-module.

If $N_1\longrightarrow N_2$ is injective (resp. surjective), is the induced map $M\otimes_R N_1\longrightarrow M\otimes N_2$ necessarily injective (resp. surjective)?

I really do not know how to prove and cannot give counterexamples.

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Injectivity is definitely not preserved. Take $N_1 = N_2 = \mathbb{Z}$ and and take the map from $N_1$ to $N_2$ given by multiplication by $2$. This map is injective, but after tensoring with $\mathbb{Z}/2\mathbb{Z}$, it is not.

Surjectivity is preserved.

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