Lemma 47.4.2. Let $R$ be a ring and let $M$ be an $R$-module. If a projective cover of $M$ exists, then it is unique up to isomorphism.

Proof. Let $P \to M$ and $P' \to M$ be projective covers. Because $P$ is a projective $R$-module and $P' \to M$ is surjective, we can find an $R$-module map $\alpha : P \to P'$ compatible with the maps to $M$. Since $P' \to M$ is essential, we see that $\alpha$ is surjective. As $P'$ is a projective $R$-module we can choose a direct sum decomposition $P = \mathop{\mathrm{Ker}}(\alpha ) \oplus P'$. Since $P' \to M$ is surjective and since $P \to M$ is essential we conclude that $\mathop{\mathrm{Ker}}(\alpha )$ is zero as desired. $\square$

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