Proposition 4

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

Algebra, VIII, p. 175, § 10. Modules over an Artinian Ring

Every module over a left Artinian ring has a projective cover.

Let A be a left Artinian ring, and let M be a module over A. The radical $\mathfrak{r}$ of A is a nilpotent two-sided ideal, and the ring $A/\mathfrak{r}$ is semisimple (VIII, p. 173, Proposition 1). It follows that the $A/\mathfrak{r}$ -module $M/\mathfrak{r}M$ is projective. The assertion then follows from Proposition 11 of VIII, p. 164.

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

Corollary, § 10
Proposition 5, § 10
§ 11. Grothendieck Groups

Cites

VIII, p. 173, Proposition 1
Proposition 11 of VIII, p. 164