Corollary

Tag 009O, permanent. Cite it as [Bourbaki, Tag 009O].

Algebra, VIII, p. 144, § 8. Semisimple Rings

Let M be a module over the semisimple ring A. The following properties are equivalent:

(i) The A-module M is faithful.

(ii) The support of M is equal to $\mathscr{S}$ .

(iii) The A-module M is generating.

Indeed, saying that M is faithful means that its annihilator is reduced to 0, and M is generating if and only if its trace is equal to A (VIII, p. 80, Theorem 1).

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

VIII, p. 80, Theorem 1