Exercise 20, § 2

(a) Let A be a ring admitting a field of left fractions K (I, § 9, Exercise 15). Show that K, considered as a left A-module, is an injective envelope of $A_s$. (Apply criterion (α) of Exercise 18(d) and criterion (δ) of Exercise 11(a) noting that if $f : a \to K$ is an A-linear mapping of a left ideal $a$ of A into K, the element $x^{-1} f(x)$ for $x \in a, x \neq 0$ (the inverse being taken in K) does not depend on $x \in a$.)

(b) Let $U_p$ be the sub-$\mathbf{Z}$-module of $\mathbf{Q}/\mathbf{Z}$ consisting of the canonical images of the rational numbers of the form $k/p^n$, where $p$ is a given prime number, $k \in \mathbf{Z}$ and $n \in \mathbf{N}$ (II, § 1, Exercise 12(d)). Show that $U_p$ is an injective envelope of each of the $\mathbf{Z}$-modules $p^{-n}\mathbf{Z}/\mathbf{Z}$ isomorphic to $\mathbf{Z}/p^n\mathbf{Z}$. Show that there exist automorphisms of $U_p$ distinct from the identity automorphism leaving invariant the elements of a submodule $p^{-n}\mathbf{Z}/\mathbf{Z}$.