On Injective and Divisible Modules

Abstract

An R-module is one of several equivalent statements of injective modules that we will be discussing , including concepts dealing with ideals of rings and homomorphism modules. Divisible modules over general rings are considered and a general notion of divisibility is defined. In order to study these divisible modules we generalize the notion of injectivity. One consequence is that rings for which every principal right ideal is projective can be char acterized. Since this thesis is a synopsis, the research gathered is scattered throughout the paper (Head, 1974), (Hungerford, 1974), (Lam, 1999), and (Sharpe and Vamos, 1972). We discuss how these two ideas (injectivity and divisibility)compare with each other in general rings, as well as special ones such as generalizations of injective Modules , characterization of n-injective and n-divisible Modules. In this thesis, we introduce the relationship between the divisible module and the injective module. Also, we obtain some results about them such as every injective R-module is divisible but the converse is not.