Corollary 1
Tag 00C6, permanent. Cite it as [Bourbaki, Tag 00C6].
Algebra, VIII, p. 173, § 10. Modules over an Artinian Ring
A ring is semisimple if and only if it is left Artinian and has no nilpotent two-sided ideals other than 0.
A semisimple ring is left Artinian and without radical (VIII, p. 154, Proposition 4); it therefore has no nilpotent two-sided ideals other than 0. The converse follows from Proposition 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.
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Nothing.