Exercise 11, § 2

Let $A$ be a ring, and let $I$ be an $A$-module. We say that $I$ is injective if, for every exact sequence $E' \to E \to E''$ of a-linear mappings, the sequence $\mathrm{Hom}(E'', I) \to \mathrm{Hom}(E, I) \to \mathrm{Hom}(E', I)$ is exact. a) Now prove that the following properties are equivalent: α) $I$ is injective. β) For every exact sequence $0 \to E' \to E \to E'' \to 0$ of a-linear mappings, the sequence

is exact. γ) For every $A$-module $M$ and every submodule $N$ of $M$, every A-linear mapping from $N$ into $I$ extends to an A-linear mapping from $M$ into $I$.

δ) For every left ideal $a$ of $A$ and every $A$-linear mapping $f : a \to I$ there exists an element $b \in I$ such that $f(a) = ab$ for every $a \in a$. ζ) $I$ is a direct factor of every $A$-module which contains it. θ) For every $A$-module $E$ which is the sum of $I$ and a monogenic submodule, $I$ is a direct factor of $E$. (To now prove that δ) implies γ), show that property δ) implies that if $E$ is an $A$-module and $F$ is a submodule of $E$ such that $E/F$ is monogenic, then every linear mapping from $F$ into $I$ extends to a linear mapping from $E$ into $I$. Then use Zorn's theorem. To now prove that θ) implies δ), consider the $A$-module $A_s \times I = M$, the submodule $N$ of $M$ formed by the elements $(a, -f(a))$ for $a \in a$, and apply θ) to the quotient module $M/N$.) b) For a $\mathbf{Z}$-module $E$ to be injective, it is necessary and sufficient that for every $x \in E$ and every integer $n \neq 0$, there exists $y \in E$ such that $ny = x$. In particular, the $\mathbf{Z}$-modules $\mathbf{Q}$ and $\mathbf{Q}/\mathbf{Z}$ are injective.