Remark

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

Algebra, VIII, p. 200, § 11. Grothendieck Groups

Let A be a semisimple ring. Every A-module is then semisimple and projective (VIII, p. 138, Proposition 4); we therefore have the equality

\mathscr{L}\mathscr{F}(A) =\mathscr{S}\mathscr{S}(A) =\mathscr{P}(A)

(cf. No. 4 for the definitions of $\mathscr{L}\mathscr{F}(A)$ and $\mathscr{S}\mathscr{S}(A)$ ). We therefore have $K_0(A) = R(A)$ by the definitions of these Grothendieck groups.

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

Nothing.

Cites

VIII, p. 138, Proposition 4