In this chapter we study the structure of the endomorphism rings of continuous and quasi–injective modules. Though many of the basic lemmas hold for quasi–continuous modules, the endomorphism ring of a continuous module M possesses some crucial properties which fail we only assume that M is quasi–continuous.

As an application of these results, in conjunction with some theorems proved in previous chapters, we show that continuous modules have the exchange property.

Beyond these facts involving the endomorphism ring, we will discuss a few other properties of continuous modules, which do not generally hold for quasi–continuous modules.

ENDOMORPHISM RINGS

Throughout this section, S will denote the endomorphism ring of a module M, J the Jacobson radical of S, Δ = {α ∈ S : Ker α ≤e M} and S/Δ.

The following lemma, whose proof is straightforward, will be used freely in this section.

Lemma 3.1. Let A be a submodule of M, α ∈ S, and e an idempotent of S. Then:

1. If A ≤e M, then eA ≤e eM;

2. αM ≤ eM if and only if αS ≤ eS.

Lemma 3.2For an arbitrary module M,

1. Δ is an ideal; and

2. if {ei : i ∈ I} is a family of idempotents of S which are orthogonal modulo Δ, then eiM is direct.

PROOF. (1) Let a, b ∈ Δ and α ∈ S. Then Ker a ≤e M and Ker b ≤e M.