Remark 1
Tag 0083, permanent. Cite it as [Bourbaki, Tag 0083].
Algebra, VIII, p. 120, § 7. Simple Rings
Recall that by Theorem 1 of II, §9, No. 1, p. 115, the following properties are equivalent:
(i) The A-module is simple.
(ii) The ring A is not reduced to 0, and there are no left ideals of A distinct from 0 or A.
(iii) The ring A is a field.
Consequently, by Proposition 1 (condition (ii)), commutative simple rings are nothing but commutative fields.
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.