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

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

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

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.
$\square$

## Comments (2)

Comment #2471 by Raymond Cheng on

Comment #2504 by Johan on