Some generalizations of injectivity
Chapter 1 covers the background necessary for what follows. In particular, general properties of injectivity and some of its well-known generalizations are stated. Chapter 2 is concerned with two generalizations of injectivity, namely near and essential injectivity. These concepts, together with the notion of the exchange property, prove to be a key tool in obtaining characterizations of when the direct sum of extending modules is extending. We find sufficient conditions for a direct sum of two extending modules to be extending, generalizing several known results. We characterize when the direct sum of an extending module and an injective module is extending and when the direct sum of an extending module with the finite exchange property and a semisimple module is extending. We also characterize when the direct sum of a uniform-extending module and a semisimple module is uniform-extending and, in consequence, we prove that, for a right Noetherian ring R, an extending right R-module M1 and a semisimple right R-module M2, the right R-module M1 M2 is extending if and only if M2 is M1/Soc(M1)-injective. Chapter 3 deals with the class of self-c-injective modules, that can be characterised by the lifting of homomorphisms from closed submodules to the module itself. We prove general properties of self-c-injective modules and find sufficient conditions for a direct sum of two self-c-injective to be self-c-injective. We also look at self-cu-injective modules, i.e. modules M such that every homomorphism from a closed uniform submodule to M can be lifted to M itself. We prove that every self-c-injective free module over a commutative domain that is not a field is finitely generated and then proceed to consider torsion-free modules over commutative domains, as was done for extending modules in . We also characterize when, over a principal ideal domain, the direct sum of a torsion-free injective module and a cyclic torsion module is self-cu-injective.