Continuous and Discrete Modules

Mohamed, Saad H.; Müller, Bruno J.

doi:10.1017/cbo9780511600692

Cited by 559 publications

(314 citation statements)

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“…In first case a = Z and in second case a = Z p n . So Z p must be Z-injective or Z p n -injective by [8,Proposition 1.4]. But the homomorphism f : pZ → Z p with f (p) = 1 cannot be extended to g : Z → Z p since otherwise 1 = f (p) = g(p) = pg(1) = 0 and Z p is isomorphic to the subgroup p n−1 of Z p n which is not a direct summand of Z p n .…”

Section: Poor Abelian Groupsmentioning

confidence: 99%

Poor and pi-poor Abelian groups

Alizade

Büyükaşık

2016

Communications in Algebra

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Abstract. In this paper, poor abelian groups are characterized. It is proved that an abelian group is poor if and only if its torsion part contains a direct summand isomorphic to ⊕ p∈P Zp, where P is the set of prime integers. We also prove that pi-poor abelian groups exist. Namely, it is proved that the direct sum of U (N) , where U ranges over all nonisomorphic uniform abelian groups, is pi-poor. Moreover, for a pi-poor abelian group M , it is shown that M can not be torsion, and each p-primary component of M is unbounded. Finally, we show that there are pi-poor groups which are not poor, and vise versa.

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Section: Poor Abelian Groupsmentioning

confidence: 99%

Poor and pi-poor Abelian groups

Alizade

Büyükaşık

2016

Communications in Algebra

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“…Quoting Kasch [17], "the concepts of projective and injective modules are among the most important fundamental concepts of the theory of rings and modules". For various generalizations of injectivity and projectivity, consult [19] (cf. also [26]).…”

Section: Scs Modulesmentioning

confidence: 99%

A Generalization of Injectivity for Modules Over a Unitary Ring

Ming¹

2006

Demonstratio Mathematica

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Abstract. In this note, we consider certain generalizations of injectivity and pinjectivity in connection with von Neumann regular rings, self-injective regular rings, II-regular rings, semi-simple Artinian and simple Artinian rings.A generalization of quasi-injective modules, noted SCS modules, is introduced. It is proved that A is a left self injective regular ring if, and only if, A is a left p-injective left non-singular left SCS ring. SCS rings are used to characterize simple Artinian rings. A generalization of p-injective modules, noted WGP-injective is used to study II-regular rings. If A is a right p.p. right WGP-injective ring, then A is II-regular. If A is a semi-prime ring whose simple left modules are either WGP-injective or projective, then the centre of A is von Neumann regular. Left Artinian rings are characterized as left Noetherian rings whose prime factor rings are left WGP-injective. Also, A is a left WGP-injective ring if and only if for any a 6 A, there exists a positive integer n such that a n A is a right anihilator. (Here a n may be zero.) IntroductionThroughout, A denotes an associative ring with identity and A-modules are unitial, J, Y, Z will stand respectively for the Jacobson radical, the right singular ideal and the left singular ideal of A. A is called respectively semiprimitive ( (a) Following Faith [7], A is called VNR if 11 A is a von Neumann regular ring". Recall that:

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“…Furthermore, in 1984, Okado showed the following result: a ring R is right noetherian if and only if every extending R-module has an indecomposable decomposition (cf. [8]). …”

Section: Introductionmentioning

confidence: 99%

“…Dually, in 1996, Oshiro-Rizvi proved that discrete modules have the exchange property and, for quasi-discrete modules, the finite exchange property implies the exchange property (cf. [1,4,8,9]). …”

Section: Introductionmentioning

confidence: 99%

On the Decomposition of Extending Lifting Modules

Chang¹,

Shin²

2009

Bulletin of the Korean Mathematical Society

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Abstract. In 1984, Oshiro [11] has studied the decomposition of continuous lifting modules. He obtained the following: every continuous lifting module has an indecomposable decomposition. In this paper, we study extending lifting modules. We show that every extending lifting module has an indecomposable decomposition. This result is an expansion of Oshiro's result mentioned above. And we consider some application of this result.

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Continuous and Discrete Modules

Cited by 559 publications

References 0 publications

Poor and pi-poor Abelian groups

Poor and pi-poor Abelian groups

A Generalization of Injectivity for Modules Over a Unitary Ring

On the Decomposition of Extending Lifting Modules

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