Proposition 4
Tag 00AB, permanent. Cite it as [Bourbaki, Tag 00AB].
Algebra, VIII, p. 154, § 9. Radical
A ring A is semisimple if and only if it is left Artinian and without radical.
This follows from Proposition 3, b) applied to the A-module .
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.