Chapter 2 - Quasi–Continuous Modules

Summary

In this chapter we discuss generalizations of the notion of continuous rings studied by von Neumann [36] and Utumi [65] to modules. Such modules are also generalizations of (quasi–) injective modules.

BASIC PROPERTIES

Proposition 2.1. Any (quasi–)injective module M satisfies the following two conditions: (C₁). Every submodule of M is essential in a summand of M; (C₂) If a submodule A of M is isomorphic to a summand of M, then A is a summand of M.

PROOF. Let N ≤ M and write E(M) = E₁ ⊕ E₂ where E₁ = E(N). The quasi-injectivity of M implies, by Corollary 1.14, that M = M ∩ E₁ ⊕ M ∩ E₂; and it is clear that N ≤ ^e M ∩ E₁. Hence (C₁) holds.

Let M′ M be a monomorphism with M′ c^⊕ M. Since M is M–injective, M′ is M–injective by Proposition 1.6. Then f splits by Lemma 1.2; thus (C₂) holds.

Proposition 2.2. If a module M has (C₂), then it satisfies the following condition: (C₃) If M₁and M₂are summands of M such that M₁ ∩ M₂ = 0, then M₁ ⊕ M₂is a summand of M.

PROOF. Write M = M₁ ⊕ M₁^* and let π denote the projection M₁^* ⊕ M₁ →> M^*₁. Then M₁ ⊕ M₂ = M₁ ⊕ πM₂. Since π|_M2 is a monomorphism, we get πM₂ C^⊕ M by (C₂). As πM₂ ≤ M₁₁, M₁ ⊕ πM₂ C^⊕ M.

Definition 2.3. A module M is called continuous if it has (C₁) and (C₂); M is called quasi–continuous if it has (C₁) and (C₃)