Exercise 3, § 6

Let A be a ring and G an abelian group denoted additively. A mapping $T : A\rightarrow G$ is called tracial if it is a homomorphism of additive groups and we have $T(ab) = T(ba)$ for every $a, b$ in A.

a) Let $T : A\rightarrow G$ be a tracial mapping, and let P be a right A-module that is projective and finitely generated. Prove that there exists a unique tracial mapping $T_P:$ End$_A(P\oplus A_d)\rightarrow G$ whose restriction to A = End$_A(A_d)$ is T.

b) Let A and B be Morita equivalent rings and G an abelian group. Construct a bijective correspondence between tracial mappings from A to G and tracial mappings from B to G.