Exercise 5, § 11

Let A be a ring. Denote by $\mathscr{I}(A)$ the set of pairs $(n, p)$ with $n$ an integer and $p$ an idempotent in $M_n(A)$. Define a law of addition on $\mathscr{I}(A)$ by setting $(m, p) + (n, q) = (m+n,\begin{pmatrix} p & 0 \\ 0 & q \end{pmatrix}$ ).

a) Let $\mathscr{P}(A^o)$ be the monoid of classes of finitely generated projective right A-modules. Let $\varphi :\mathscr{I}(A)\rightarrow \mathscr{P}(A^o)$ be the mapping defined by $\varphi (n, p) =$ cl(Im $p$). Prove that $\varphi$ is a surjective homomorphism and that elements $(m, p)$ and $(n, q)$ of $\mathscr{I}(A)$ have the same image by $\varphi$ if and only if there exist $a\in M_{m,n}(A)$ and $b\in$ $M_{n,m}(A)$ such that $ab=p$ and $ba=q$.

b) Deduce that the groups $K_0(A)$ and $K_0(A^o)$ are isomorphic.