The Stacks project

Definition 46.8.1. Let $A$ be a ring.

1. An $A$-module $P$ is said to be pure projective if for every universally exact sequence $0 \to K \to M \to N \to 0$ of $A$-module the sequence $0 \to \mathop{\mathrm{Hom}}\nolimits _ A(P, K) \to \mathop{\mathrm{Hom}}\nolimits _ A(P, M) \to \mathop{\mathrm{Hom}}\nolimits _ A(P, N) \to 0$ is exact.

2. An $A$-module $I$ is said to be pure injective if for every universally exact sequence $0 \to K \to M \to N \to 0$ of $A$-module the sequence $0 \to \mathop{\mathrm{Hom}}\nolimits _ A(N, I) \to \mathop{\mathrm{Hom}}\nolimits _ A(M, I) \to \mathop{\mathrm{Hom}}\nolimits _ A(K, I) \to 0$ is exact.