Exercise 5, § 6

For a ring R, denote by $\mathbf{M}_{\infty}(R)$ the pseudoring of matrices of type $\mathbf{N}\times \mathbf{N}$ with entries in R having only finitely many nonzero entries. Prove that the rings A and B are Morita equivalent if and only if the pseudorings $\mathbf{M}_{\infty}(A)$ and $\mathbf{M}_{\infty}(B)$ are isomorphic. (For an invertible $(A$, B)-module P, consider the image of $\mathbf{M}_{\infty}(A)$ under the isomorphism End$_A(A^{(\mathbf{N})}_s)\rightarrow$ End$_A(P^{(\mathbf{N})})$ constructed in Exercise 12 of VIII, p. 92. Given an isomorphism from $\mathbf{M}_{\infty}(A)$ to $\mathbf{M}_{\infty}$(B), consider the image in $\mathbf{M}_{\infty}(B)$ of the idempotent $E_{11}$ in $\mathbf{M}_{\infty}(A).)$