r-injective modules

Let M and N be two R-modules. If for each g: K→N R-module monomorphism satisfying RadN≤g(K) and every f: K→M R-module homomorphism, there exists at least one h: N→M R-module homomorphism such that hog=f, then M is called a radical N-injective (briefly, r-N-injective) module. If the module M is radical T-injective for every R-module T, then M is called a radical injective (briefly, r-injective) module. In this thesis, the concept of r-injective module is considered and some properties related to this concept are investigated. All modules are unitary modules over the ring with unity, in this study. It is clear that for an R-module N, every N-injective module is r-N-injective. But the converse is not true in general. Let M and N be two R-modules, and let every submodule K of N be a direct summand of K+RadN. In this case, M is r-N-injective if and only if M is N-injective. Let M and N be two R-modules. In this case, M is r-N-injective if and only if for every K≤N with RadN≤K and every f: K→M R-module homomorphism, there exists an R-module homomorphism h: NM such that hoi=f with an inclusion map i: K→N. Let N be an R-module, K≤N and RadN≤K. If K is r-N-injective, then K is a direct summand of N. In this thesis, it is shown that the product and the direct sum of r-N-injective modules are also r-N-injective, given that N is an R-module. Furthermore, it is shown that every direct summand of an r-N-injective module is also r-N-injective. Let N be an R-module. In this case, every R-module is r-N-injective if and only if every submodule of N containing RadN is a direct summand of N. Let R be a ring. In this case, every R-module is r-injective if and only if every R-module is injective. Four examples related to the topic are provided at the end of the thesis. Keywords: Small submodule, Radical, Exact sequence, Injective module.