Exercise 9, § 11

Let A be a pseudoring, that is, an associative not necessarily unital $\mathbf{Z}$-algebra, and $\widetilde{A}$ the $\mathbf{Z}$-algebra deduced from A by adjoining a unit element (III, §1, No. 2, p. 431); the set $\widetilde{A}$ is equal to $\mathbf{Z}\times A$. Denote the homomorphism $(n, a)\mapsto n$ by $\varepsilon :\widetilde{A}\rightarrow \mathbf{Z}$, and denote the kernel of the homomorphism $\varepsilon^*: K_0(\widetilde{A})\rightarrow K_0(\mathbf{Z})$ associated with $\varepsilon$ by $K_0(A)$.

a) Prove that if A is unital, then $\widetilde{A}$ is isomorphic to the product ring $\mathbf{Z}\times A$ and the definition of $K_0(A)$ is compatible with that given in No.8.

b) Let A and B be pseudorings and $f: A\rightarrow B$ a homomorphism. Let $\widetilde{f}:\widetilde{A}\rightarrow \widetilde{B}$ be the ring homomorphism defined by $\widetilde{f}((n, a)) = (n, f(a))$. Prove that $\widetilde{f}^*$ maps $K_0(A)$ into $K_0(B)$. Denote the homomorphism deduced from $\widetilde{f}^*$ by $f^*: K_0(A)\rightarrow K_0(B)$. c) Suppose that $f$ is surjective; denote its kernel by $\mathfrak{a}$ and the canonical injection of $\mathfrak{a}$ into A by $i$. Show that the sequence

is exact. If, moreover, there exists a homomorphism of pseudorings $s: B\rightarrow A$ such that $f\circ s=$ Id$_B$, then $i^*$ is injective and $f^*$ is surjective (apply Exercise 8 by first taking $(A,A,B)$ and then $(D,\mathbf{Z},A)$ for the triple $(A,B,C)$).

d) Prove that the conclusions of Exercises 7 and 8 extend to the case of pseudorings. e) Denote by $\mathbf{M}_{\infty}(A)$ the pseudoring of $\mathbf{N}\times \mathbf{N}$ matrices with entries in A having only finitely many nonzero entries. Let $f: A\rightarrow \mathbf{M}_{\infty}(A)$ be the mapping that sends $a\in A$ to the matrix $(b_{ij})$ with $b_{00}=a$ and $b_{ij}= 0$ if $(i, j)\not= (0,0)$. Prove that $f^*$ is an isomorphism from $K_0(A)$ to $K_0(\mathbf{M}_{\infty}(A))$ (use Exercises 4 and 7).