Mathematics / Matematik
Permanent URI for this collectionhttps://hdl.handle.net/11147/8
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Article Citation - WoS: 3Citation - Scopus: 3Dual Kasch rings(World Scientific Publishing, 2023) Lomp, Christian; Büyükaşık, Engin; Yurtsever, Haydar BaranIt is well known that a ring R is right Kasch if each simple right R-module embeds in a projective right R-module. In this paper we study the dual notion and call a ring R right dual Kasch if each simple right R-module is a homomorphic image of an injective right R-module. We prove that R is right dual Kasch if and only if every finitely generated projective right R-module is coclosed in its injective hull. Typical examples of dual Kasch rings are self-injective rings, V-rings and commutative perfect rings. Skew group rings of dual Kasch rings by finite groups are dual Kasch if the order of the group is invertible. Many examples are given to separate the notion of Kasch and dual Kasch rings. It is shown that commutative Kasch rings are dual Kasch, and a commutative ring with finite Goldie dimension is dual Kasch if and only if it is a classical ring (i.e. every element is a zero divisor or invertible). We obtain that, for a field k, a finite dimensional k-algebra is right dual Kasch if and only if it is left Kasch. We also discuss the rings over which every simple right module is a homomorphic image of its injective hull, and these rings are termed strongly dual Kasch.Article Citation - WoS: 14Citation - Scopus: 12Injective modules over down-up algebras(Cambridge University Press, 2010) Carvalho, Paula A.A.B.; Lomp, Christian; Pusat, DilekThe purpose of this paper is to study finiteness conditions on injective hulls of simple modules over Noetherian down-up algebras. We will show that the Noetherian down-up algebras A(α, β, γ) which are fully bounded are precisely those which are module-finite over a central subalgebra. We show that injective hulls of simple A(α, β, γ)-modules are locally Artinian provided the roots of X2 − αX − β are distinct roots of unity or both equal to 1.Article Citation - WoS: 19Citation - Scopus: 15When Δ-Semiperfect Rings Are Semiperfect(TUBITAK, 2010) Büyükaşık, Engin; Lomp, ChristianZhou defined δ -semiperfect rings as a proper generalization of semiperfect rings. The purpose of this paper is to discuss relative notions of supplemented modules and to show that the semiperfect rings are precisely the semilocal rings which are δ -supplemented. Module theoretic version of our results are obtained. © TÜBİTAK.Article Citation - WoS: 16Citation - Scopus: 14Rings Whose Modules Are Weakly Supplemented Are Perfect. Applications To Certain Ring Extensions(Mathematica Scandinavica, 2009) Büyükaşık, Engin; Lomp, ChristianIn this note we show that a ring R is left perfect if and only if every left R-module is weakly supplemented if and only if R is semilocal and the radical of the countably infinite free left R-module has a weak supplement.