Definition 2
Tag 00PD, permanent. Cite it as [Bourbaki, Tag 00PD].
Algebra, VIII, p. 440, Appendix 1. Algebras without Unit Element
We say that a pseudomodule M over A is simple if we have AM and if 0 and M are the only sub-pseudomodules of M.
This corresponds to saying that the -module M is simple and that its annihilator does not contain A. When A is a ring and M is an A-module, we have AM = M, so Definition 2 coincides with Definition 1 of VIII, p. 45.
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.