Phd Degree / Doktora
Permanent URI for this collectionhttps://hdl.handle.net/11147/2869
Browse
2 results
Search Results
Doctoral Thesis When Certain Relative Projectivity and Injectivity Conditions Imply the Global Projectivity and Injectivity(Izmir Institute of Technology, 2022) Benli Göral, Sinem; Büyükaşık, EnginA right R-module M is called R-projective provided that it is projective relative to the right R-module RR. One of the parts of this thesis deals with the rings whose all nonsingular right modules are R-projective. For a right nonsingular ring R, we prove that RR is of finite Goldie rank and all nonsingular right R-modules are R-projective if and only if R is right finitely Σ-CS and flat right R-modules are R-projective. Then, R-projectivity of the class of nonsingular injective right modules is also considered. Over right nonsingular rings of finite right Goldie rank, it is shown that R-projectivity of nonsingular injective right modules is equivalent to R-projectivity of the injective hull E(RR). As a second goal, we deal with simple-injective modules. For a right module M, we prove that M is simple-injective if and only if M is min-N-injective for every cyclic right module N. The rings whose simple-injective right modules are injective are exactly the right Artinian rings. A right Noetherian ring is right Artinian if and only if every cyclic simple-injective right module is injective. The ring is quasi-Frobenius if and only if simple-injective right modules are projective. For a commutative Noetherian ring R, we prove that every finitely generated simple-injective R-module is projective if and only if R = A × B, where A is quasi-Frobenius and B is hereditary. An abelian group is simpleinjective if and only if its torsion part is injective.Doctoral Thesis Homological Objects of Proper Classes Generated by Simple Modules(Izmir Institute of Technology, 2014) Durğun, Yılmaz; Büyükaşık, EnginThe main purpose of this thesis is to study some classes of modules determined by neat, coneat and s-pure submodules. A right R-module M is called neat-flat (resp. coneat-flat) if the kernel of any epimorphism Y → M → 0 is neat (resp. coneat) in Y. A right R-module M is said to be absolutely s-pure if it is s-pure in every extension of it. If R is a commutative Noetherian ring, then R is C-ring if and only if coneat-flat modules are flat. A commutative ring R is perfect if and only if coneat-flat modules are projective. R is a right Σ -CS ring if and only if neat-flat right R-modules are projective. R is a right Kasch ring if and only if injective right R-modules are neat-flat if and only if the injective hull of every simple right R-module is neat-flat. If R is a right N-ring, then R is right Σ -CS ring if and only if pure-injective neat-flat right R-modules are projective if and only if absolutely s-pure left R-modules are injective and R is right perfect. A domain R is Dedekind if and only if absolutely s-pure modules are injective. It is proven that, for a commutative Noetherian ring R, (1) neat-flat modules are flat if and only if absolutely s-pure modules are absolutely pure if and only if R A × B, wherein A is QF-ring and B is hereditary; (2) neat-flat modules are absolutely s-pure if and only if absolutely s-pure modules are neat-flat if and only if R A × B, wherein A is QF-ring and B is Artinian with J2(B) = 0.