Injectivity and Some of Its Generalizations

Abstract

Conditions which generalize injective modules and which relate to this notion are presented in this chapter. The notions of extending, quasi-continuous and continuous modules are discussed and a number of their properties are included. It is known that direct summands of modules satisfying either of (C1), (C2) or (C3) conditions, of (quasi-)injective modules, or of (quasi-)continuous modules inherit these respective properties. On the other hand, these classes of modules are generally not closed under direct sums. One focus of this chapter is to discuss conditions which ensure that such classes of modules are also closed under direct sums. Applications of these notions, which include decomposition theorems, are also considered. As a natural generalization of the extending property, the notion of an FI-extending module (i.e., a module for which every fully invariant submodule is essential in a direct summand) is presented. A longstanding open problem is the precise characterization of when a direct sum of extending modules is extending. Using the FI-extending property, one can see that an arbitrary direct sum of extending modules satisfies at least the extending property for its fully invariant submodules without any additional conditions. The closely related notion of strongly FI-extending modules is also introduced. Properties of FI-extending and strongly FI-extending modules are discussed.