Remark 1

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

Algebra, VIII, p. 237, § 13. Absolutely Semisimple Algebras

Let $e=\sum^r_{i=1}a_i\otimes a'_i$ be an element of B. The conditions in item (ii) of Proposition 5 translate to the formulas

\sum_{i=1}^ra_ia'_i= 1 \tag{2}

(3) $\sum_{i=1}^raa_i\otimes a'_i=\sum_{i=1}^ra_i\otimes a'_ia$ for every $a\in A$ .

When they are satisfied, $e$ is an idempotent in B. Indeed, we then have the relations

e^2=\sum_{i=1}^ra_iea'_i=\sum_{i=1}^rea_ia'_i=e

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

Proposition 7, § 13

Cites

Proposition 5