Exercise 9, § 2

Let $E$ be a finitely presented left $A$-module. (a) Show that for every family $(F_i)_{i \in I}$ of right $A$-modules, the canonical homomorphism $E \otimes_A \left( \prod_{i \in I} F_i \right) \to \prod_{L \in I} (E \otimes_A F_i)$ (Algebra, Chapter II, § 3, no. 7) is bijective. (b) Let $(G_\alpha, \phi_{\beta\alpha})$ be a direct system of left $A$-modules; show that the Canonical homomorphism

is bijective.