Injective Modules And Divisible Groups

An R-module M is injective provided that for every R-monomorphism g from R-modules A to B, any R-homomorphism f from A to M can be extended to an R-homomorphism h from B to M such that hg = [equals] f. That is one of several equivalent statements of injective modules that we will be discussing, including concepts dealing with ideals of rings, homomorphism modules, short exact sequences, and splitting sequences. A divisible group G is defined when for every element x of G and every nonzero integer n, there exists y in G such that x = [equals] ny. We will see how these two ideas (injectivity and divisibility) compare with each other in general rings, as well as special ones such as Noetherian, Dedekind, and Semi-simple. Since this thesis is a synopsis, the research gathered is scattered throughout the paper (Head, 1974), (Hungerford, 1974), (Lam, 1999), (Rotman, 1995), (Rotman, 1979), and (Sharpe and Vamos, 1972).