Düzenlilik ve homomorfizmaların genişlemeleri

This thesis consists of two main parts. The first part is concerned with neat submodules over commutative rings and the second part is about neat homomorphisms and extensions of homomorphisms over an arbitrary ring. Let R be a commutative ring with identity. A short exact sequence E of R-modules is said to be neat (respectively, P-pure) if the sequence HomR(S, E) (respectively, S ⊗R E) is exact for every simple R-module S. We generalized Fuchs' characterization of integral domains where neatness and P-purity coincide to commutative rings. We show that every maximal ideal of R is finitely generated and projective if and only if R has projective socle and neatness and P-purity coincide. We prove that neatness and P-purity coincide if and only if every maximal ideal of R is finitely generated and the unique maximal ideal PP of the local ring RP is principal for every maximal ideal P of R. This result is proved firstly over commutative local rings and then using localization over any commutative ring. The Auslander-Bridger transpose of simple modules is used for the passage between proper classes of short exact sequences of modules that are projectively generated and these that are flatly generated by a set of finitely presented modules. The second part of this thesis presents our work with N. Er in a detailed form; we investigate the following properties which are motivated by our discussion on neatness: A ring R is said to satisfy the property (P) if all nonzero R-module epimorphisms with cyclic domains are neat; it is said to satify (Q) if every non-injective R-module is a test for injectivity by neatness. A large class of rings including local perfect rings and those that satisfy (Q) share a common property: any two noninjective modules admit a nonzero homomorphism between them (property (T)). All three classes of rings are characterized and various examples are provided.