Exercise 6, § 11

Let A be a ring, G an abelian group, and $T : A\rightarrow G$ a tracial mapping (VIII, p. 115, Exercise 3). For any finitely generated projective right A-module P, denote by

$T_P:$ End$_A(P\oplus A_d)\longrightarrow G$

the unique tracial mapping that extends T (loc. cit.) and by $\pi_P$ the projector of End$_A(P\oplus A_d)$ with image P and kernel $A_d$. Construct a group homomorphism $\widetilde{T} : K_0(A)\rightarrow G$ such that $\widetilde{T}([P]) = T_P(\pi_P)$ for every finitely generated projective right A-module P.