Corollary

Let J be an injective A-module such that Ass_A(J) = {p}.

a) The canonical mapping J → A_p ⊗_A J is bijective.

b) Let E denote the A/p-module Hom_A(A/p, J). There exists on E a unique structure of κ(p)-vector space extending its structure of A/p-module; the A-module J is isomorphic to I([E:κ(p)]).

Indeed, J is isomorphic to an A-module I^{(c)}, where c is a suitable cardinal (No. 1, th. 1). The corollary follows from the proposition when J = I, and the general case is deduced from it at once.