Exercise 13, § 2

a) Let $A$ be a ring, $I$ an injective $A$-module (exerc. 11), such that, for every nonzero monogenic $A$-module $E$, there exists a nonzero $A$-homomorphism from $E$ into $I$. Let $M$ be an $A$-module, $N$ a submodule of $M$, $\tilde{N}$ the set of the $u \in \mathrm{Hom}_A(M, I)$ such that $u(x) = 0$ in $N$. Now prove that for every $y \notin N$, there exists a $u \in \tilde{N}$ such that $u(y) \neq 0$. b) Let $A$ be a ring. For every $(A, A)$-bimodule $T$, and every left (resp. right) $A$-module $M$, $\mathrm{Hom}_A(M, T)$ is canonically endowed with a right (resp. left) $A$-module structure coming from that of $T$, and there is a canonical $A$-homomorphism $c_{M, T} : M \to \mathrm{Hom}_A(\mathrm{Hom}_A(M, T), T)$ such that $c_{M, T}(x)$ is the $A$-homomorphism $u \mapsto u(x)$. Now prove that if $I$ is an $(A, A)$-bimodule which, as a left $A$-module, satisfies the conditions of a), then the canonical homomorphism $c_{M, I}$ is injective for every left $A$-module $M$. c) Suppose that $I$ is an $(A, A)$-bimodule, that it is injective as a left $A$-module and right $A$-module, and that for every monogenic $A$-module (left or right) $E \neq 0$, there exists a nonzero $A$-homomorphism from $E$ into $I$. Now prove that under these conditions, if $P$ is a projective left $A$-module (resp. right), $\mathrm{Hom}_A(P, I)$ is an injective right $A$-module (resp. left). (Suppose that $P$ is a projective left $A$-module. Observe that for every submodule $N'$ of a right $A$-module $N$, $\mathrm{Hom}_A(N', I)$ is isomorphic to a quotient module of $\mathrm{Hom}_A(N, I)$ and apply b) to the right module $N$ and the left module $P$.) d) Let $C$ be a commutative ring, $A$ a $C$-algebra, $I$ an injective $C$-module such that for every monogenic $C$-module $E \neq \{0\}$ there exists a nonzero $C$-homomorphism from $E$ into $I$. For every $A$-module $M$, $\mathrm{Hom}_C(M, I)$ is an $A$-module; now prove that for every projective $A$-module $P$, $\mathrm{Hom}_C(P, I)$ is an injective $A$-module (same reasoning as in c)).