Definition 10.82.1. Let $f: M \to N$ be a map of $R$-modules. Then $f$ is called *universally injective* if for every $R$-module $Q$, the map $f \otimes _ R \text{id}_ Q: M \otimes _ R Q \to N \otimes _ R Q$ is injective. A sequence $0 \to M_1 \to M_2 \to M_3 \to 0$ of $R$-modules is called *universally exact* if it is exact and $M_1 \to M_2$ is universally injective.

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