Lemma 10.82.12. Let $R \to S$ be a ring map. Let $M \to M'$ be a map of $S$-modules. The following are equivalent

$M \to M'$ is universally injective as a map of $R$-modules,

for each prime $\mathfrak q$ of $S$ the map $M_{\mathfrak q} \to M'_{\mathfrak q}$ is universally injective as a map of $R$-modules,

for each maximal ideal $\mathfrak m$ of $S$ the map $M_{\mathfrak m} \to M'_{\mathfrak m}$ is universally injective as a map of $R$-modules,

for each prime $\mathfrak q$ of $S$ the map $M_{\mathfrak q} \to M'_{\mathfrak q}$ is universally injective as a map of $R_{\mathfrak p}$-modules, where $\mathfrak p$ is the inverse image of $\mathfrak q$ in $R$, and

for each maximal ideal $\mathfrak m$ of $S$ the map $M_{\mathfrak m} \to M'_{\mathfrak m}$ is universally injective as a map of $R_{\mathfrak p}$-modules, where $\mathfrak p$ is the inverse image of $\mathfrak m$ in $R$.

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