Yazar "Buyukasik, Engin" seçeneğine göre listele

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    Max-projective modules
    (World Scientific Publ Co Pte Ltd, 2021) Alagoz, Yusuf; Buyukasik, Engin
    Weakening the notion of R-projectivity, a right R-module M is called max-projective provided that each homomorphism f : M -> R/I, where I is any maximal right ideal, factors through the canonical projection pi : R -> R/I. We study and investigate properties of max-projective modules. Several classes of rings whose injective modules are R-projective (respectively, max-projective) are characterized. For a commutative Noetherian ring R, we prove that injective modules are R-projective if and only if R = A x B, where A is QF and B is a small ring. If R is right hereditary and right Noetherian then, injective right modules are max-projective if and only if R = S x T, where S is a semisimple Artinian and T is a right small ring. If R is right hereditary then, injective right modules are max-projective if and only if each injective simple right module is projective. Over a right perfect ring max-projective modules are projective. We discuss the existence of non-perfect rings whose max-projective right modules are projective.
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    On max-flat and max-cotorsion modules
    (Springer, 2021) Alagoz, Yusuf; Buyukasik, Engin
    In this paper, we continue to study and investigate the homological objects related to s-pure and neat exact sequences of modules and module homomorphisms. A right module A is called max-flat if Tor(1)(R) (A, R/I) = 0 for any maximal left ideal I of R. A right module B is said to be max-cotorsion if Ext(R)(1)(A, B) = 0 for any max-flat right module A. We characterize some classes of rings such as perfect rings, max-injective rings, SF rings and max-hereditary rings by max-flat and max-cotorsion modules. We prove that every right module has a max-flat cover and max-cotorsion envelope. We show that a left perfect right max-injective ring R is QF if and only if maximal right ideals of R are finitely generated. The max-flat dimensions of modules and rings are studied in terms of right derived functors of -circle times-. Finally, we study the modules that are injective and flat relative to s-pure exact sequences.
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    On Purities Relative to Minimal Right Ideals
    (Maik Nauka/Interperiodica/Springer, 2023) Alagoz, Yusuf; Alizade, Rafail; Buyukasik, Engin; Sagbas, Selcuk
    We call a right module M weakly neat-flat if Hom(S, N) -> Hom(S, M) is surjective for any epimorphism N -> M and any simple right ideal S. A left module M is called weakly absolutely s-pure if S circle times M -> S circle times N is monic, for any monomorphism M -> N and any simple right ideal S. These notions are proper generalization of the neat-flat and the absolutely s-pure modules which are defined in the same way by considering all simple right modules of the ring, respectively. In this paper, we study some closure properties of weakly neat-flat and weakly absolutely s-pure modules, and investigate several classes of rings that are characterized via these modules. The relation between these modules and some well-known homological objects such as projective, flat, injective and absolutely pure are studied. For instance, it is proved that R is a right Kasch ring if and only if every weakly neat-flat right R-module is neat-flat (moreover if R is right min-coherent) if and only if every weakly absolutely s-pure left R-module is absolutely s-pure. The rings over which every weakly neat-flat (resp. weakly absolutely s-pure) module is injective and projective are exactly the QF rings. Finally, we study enveloping and covering properties of weakly neat-flat and weakly absolutely s-pure modules. The rings over which every simple right ideal has an epic projective envelope are characterized.
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    On simple-injective modules
    (World Scientific Publ Co Pte Ltd, 2023) Alagoz, Yusuf; Benli-Goral, Sinem; Buyukasik, Engin
    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 QF 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 x B, where A is QF and B is hereditary. An abelian group is simple-injective if and only if its torsion part is injective. We show that the notions of simple-injective, strongly simple-injective, soc-injective and strongly soc-injective coincide over the ring of integers.
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    Rings whose nonsingular right modules are R-projective
    (Charles Univ, Fac Mathematics & Physics, 2021) Alagoz, Yusuf; Benli, Sinem; Buyukasik, Engin
    A right R-module M is called R-projective provided that it is projective relative to the right R-module R-R. This paper deals with the rings whose all nonsingular right modules are R-projective. For a right nonsingular ring R, we prove that R-R is of finite Goldie rank and all nonsingular right R-modules are R-projective if and only if R is right finitely Sigma-CS and fiat 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(R-R). In this case, the injective hull E(R-R) has the decomposition E(R-R) = U-R circle plus V-R, where U is projective and Hom(V, R/I) = 0 for each right ideal I of R. Finally, we focus on the right orthogonal class N-perpendicular to of the class IV of nonsingular right modules.