5 - Injective Hulls

Summary

Injective modules may be regarded as modules that are “complete” in the following algebraic sense: Any “partial” homomorphism (from a submodule of a module B) into an injective module A can be “completed” to a “full” homomorphism (from all of B) into A. Other types of completeness often entail similar extension properties. For instance: (a) If X and Y are metric spaces with X complete, then any uniformly continuous map from a dense subspace of Y to X entends to a uniformly continuous map from Y to X; (b) if Y is a normed linear space, then any bounded linear map from a linear subspace of Y to ℝ extends to a bounded linear map from Y to ℝ; and (c) if X and Y are boolean algebras with X complete, then any boolean homomorphism from a subalgebra of Y to X extends to a boolean homomorphism from Y to X.

In topological and order-theoretic contexts, incomplete objects can be investigated by enlarging them to their completions. Following this pattern, one way to study a module A is to “complete” it to an injective module, i.e., to embed A in an injective module E, called the “injective hull” of A, in some minimal fashion. The minimality is achieved by requiring E to be an “essential extension” of A, meaning that every nonzero submodule of E has nonzero intersection with A.