Lemma 15.23.12. Let $R$ be a Noetherian ring. Let $\varphi : M \to N$ be a map of $R$-modules. Assume that for every prime $\mathfrak p$ of $R$ at least one of the following happens

1. $M_\mathfrak p \to N_\mathfrak p$ is injective, or

2. $\mathfrak p \not\in \text{Ass}(M)$.

Then $\varphi$ is injective.

Proof. Let $\mathfrak p$ be an associated prime of $\mathop{\mathrm{Ker}}(\varphi )$. Then there exists an element $x \in M_\mathfrak p$ which is in the kernel of $M_\mathfrak p \to N_\mathfrak p$ and is annihilated by $\mathfrak pR_\mathfrak p$ (Algebra, Lemma 10.63.15). This is impossible in both cases. Hence $\text{Ass}(\mathop{\mathrm{Ker}}(\varphi )) = \emptyset$ and we conclude $\mathop{\mathrm{Ker}}(\varphi ) = 0$ by Algebra, Lemma 10.63.7. $\square$

Comment #2471 by Raymond Cheng on

Third sentence: "This is impossible in all three cases." There are only two cases!

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