Corollary

Tag 00B1, permanent. Cite it as [Bourbaki, Tag 00B1].

Algebra, VIII, p. 162, § 9. Radical

If an A-module M has a projective cover, then the $A/\mathfrak{r}$ -module $M/\mathfrak{r}M$ is projective.

Indeed, if $(P, u)$ is a projective cover of M, then the $(A/\mathfrak{r}$ )-module $M/\mathfrak{r}M$ is isomorphic to $P/\mathfrak{r}P$ (Proposition 9). Since the A-module P is projective, the $(A/\mathfrak{r}$ )-module $P/\mathfrak{r}P$ is also projective.

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

Proposition 9