Corollary 2

Tag 00C7, permanent. Cite it as [Bourbaki, Tag 00C7].

Algebra, VIII, p. 173, § 10. Modules over an Artinian Ring

In a left Artinian ring A, the radical consists of the elements $x$ such that ax is nilpotent for every $a$ in A.

If A is left Artinian, then the radical is a nilpotent two-sided ideal (Proposition 1). Every left nil ideal is contained in the radical by Jacobson’s theorem (VIII, p. 156, Theorem 1). Corollary 2 follows.

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

Lemma 8, § 16
Remark 2, § 9

Cites

Proposition 1
VIII, p. 156, Theorem 1