Exercise 5, § 5

Let Γ(Z) be the gamma algebra of the Z-module Z (Exercise 2). For each p ∈ N put T^{(p)} = γ_p(1). a) Show that (T^{(p)})_{p ∈ N} is a basis of the Z-module Γ(Z). Show that there exists a unique homomorphism θ of the algebra Γ(Z) into the algebra Γ(Z) ⊗ Γ(Z) such that

for all p ∈ N. b) Let B be a commutative ring and H the set of homomorphisms of the ring Γ(Z) into the ring B. Show that for every f ∈ H the sequence f(T^{(p)}), p ∈ N is a sequence of exponential type of elements of B. Show that the mapping φ : f ↦ (f(T^{(p)}))_{p ∈ N} is a bijection of H onto $\mathscr{D}(B)$ (Exercise 1). c) For f, g ∈ H denote by f \* g the homomorphism of Γ(Z) ⊗ Γ(Z) into B defined by (f \* g)(u ⊗ v) = f(u)g(v) for any u, v ∈ Γ(Z). Verify that for any f, g ∈ H we have