Exercise 3, § 3

(a) Let $(E_\lambda)_{\lambda \in L}$ be a family of right $A$-modules and $F$ a left $A$-module. Show that if $F$ is finitely generated, the canonical mapping

is surjective. (b) Give an example of a commutative ring $A$ and an ideal $m$ of $A$ such that the canonical mapping $A^N \otimes_A (A/m) \to (A/m)^N$ is not injective (cf. § 1, Exercise 2 (b)). (c) Give an example of a commutative ring $A$ such that the canonical mapping $A^N \otimes_A A^N \to A^{N \times N}$ is not surjective (observe that for given $n$, the image of $N$ under a mapping of the form $m \mapsto \sum_{i=1}^r u_i(m)v_i(n)$, where the $u_i$ and $v_i$ belong to $A^N$, generate a finitely generated ideal of $A$). (d) Deduce from (b) and (c) an example where the canonical homomorphism (22) (no. 7) is neither injective nor surjective.