Exercise 11, § 2

Let A be a ring and I an A-module. I is called injective if, for every exact sequence $E' \to E \to E''$ of A-linear mappings, the sequence

is exact.

(a) Show 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 of N into I can be extended to an A-linear mapping of 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 all $a \in a$. (ζ) I is a direct factor of every A-module containing it. (θ) For every A-module E which is the sum of I and a monogenous submodule, I is a direct factor of E.

(To prove that (δ) implies (γ), show that property (δ) implies that if E is an A-module and F a submodule of E such that $E/F$ is monogenous, then every linear mapping of F into I can be extended to a linear mapping of E into I. Then use Zorn’s Lemma. To prove that (θ) implies (δ), consider the A-module $A_s \times I = M$, the submodule N of M consisting of 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 all $x \in E$ 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.