Exercise 13, § 2

(a) Let A be a ring and I an injective A-module (Exercise 11) such that for every non-zero monogenous A-module E, there exists a non-zero A-homomorphism of E into I. Let M be an A-module, N a submodule of M and $\tilde{N}$ the set of $u \in \operatorname{Hom}_A(M, I)$ such that $u(x) = 0$ in N. Show that for all $y \notin N$, 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 $M$, $\mathrm{Hom}_A(M, T)$ has canonically a right (resp. left) $A$-module structure deriving from that on $T$, 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 $u \mapsto u(x)$. Show 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 which is injective as a left $A$-module and a right $A$-module and that for every (left or right) monogenous $A$-module $E \neq 0$, there exists a non-zero $A$-homomorphism of $E$ into $I$. Show that under these conditions, if $P$ is a left (resp. right) projective $A$-module, $\mathrm{Hom}_A(P, I)$ is a right (resp. left) injective $A$-module. (Suppose that $P$ is a left projective $A$-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 and $I$ an injective $C$-module such that for every monogenous $C$-module $E \neq \{0\}$ there exists a non-zero $C$-homomorphism of $E$ into $I$. For every $A$-module $M$, $\mathrm{Hom}_C(M, I)$ is an $A$-module; show that for every projective $A$-module $P$, $\mathrm{Hom}_C(P, I)$ is an injective $A$-module (same argument as in $(c)$).