Definition 10.77.1 (05CE)—The Stacks project

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Definition 10.77.1. Let $R$ be a ring. An $R$ -module $P$ is projective if and only if the functor $\mathop{\mathrm{Hom}}\nolimits _ R(P, -) : \text{Mod}_ R \to \text{Mod}_ R$ is an exact functor.

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