Exercise 20, § 2

a) Let A be a ring admitting a left fractions field K (I, p. 155, Exer. 15). Show that K, considered as a left a-module of A, is an injective envelope of A_s. (Apply criterion α) of Exer. 18 d) and criterion δ) of Exer. 11 a) (II, p. 184) by noting that if f : a → K is an a-linear map of a left ideal a of A into K, the element x^{-1}f(x) for x ∈ a, x ≠ 0 (the inverse being taken in K) does not depend on x ∈ a). b) Let U_p be the sub-\mathbf{Z}-module of \mathbf{Q}/\mathbf{Z} formed by the canonical images of the rational numbers of the form k/p^n, where p is a given prime number, k ∈ \mathbf{Z} and n ∈ \mathbf{N} (II, p. 180, Exer. 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 identical automorphism, and leaving invariant the elements of a submodule p^{-n}\mathbf{Z}/\mathbf{Z}.