On 𝒯-Noncosingular Modules

Tütüncü, Derya Keskı̇n; Tribak, Rachid

doi:10.1017/s0004972709000409

Cited by 13 publications

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References 8 publications

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“…, and hence R R is not Tnoncosingular by [20,Corollary 2.7]. On the other hand, we have Z(R R ) = 0 by [4,Corollary 4.3].…”

Section: Example 23 (1) Let R Be a Right V -Ring That Is Not Semisimmentioning

confidence: 87%

“…Proof (i) ⇒ (ii) By [20,Proposition 2.3], N and N ⊕ S i are T -noncosingular modules for all i ∈ I .…”

Section: Ii) If R Is a Ring With Jac(r) = 0 Then Every Free R -Modulmentioning

confidence: 97%

“…Proof This follows from [20,Corollary 2.7] and the fact that φ ∈ Jac(S) if and only if Im φ ≪ P (see, e.g., [22, 22.2] The converse of Proposition 4.6 is not true, in general, as shown below. [20,Corollary 2.7] and [15,Proposition 2.7].…”

Section: Proposition 41 Let M Be a Quasi-discrete Module Withmentioning

confidence: 99%

“…The next example shows the existence of a T -noncosingular module that is not K -nonsingular and provides a K -nonsingular module that is not T -noncosingular. [20,Proposition 2.13], every R -module is T -noncosingular. On the other hand, from [15,Corollary 2.21] it follows that R has a module M that is not K -nonsingular.…”

Section: Some Properties Of T -Noncosingular Modulesmentioning

confidence: 99%

“…The concept of T -noncosingularity of a module was introduced and studied recently by Tütüncü and Tribak in 2009 [20] as a dual notion of the K -nonsingularity that was introduced and studied by Rizvi-Roman [14,15].…”

Section: Introductionmentioning

confidence: 99%

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Some results on T-noncosingular modules

Tribak¹

2014

Turk J Math

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The notion of T -noncosingularity of a module has been introduced and studied recently. In this article, a number of new results of this property are provided. It is shown that over a commutative semilocal ring R such that Jac(R) is a nil ideal, every T -noncosingular module is semisimple. We prove that for a perfect ring R , the class of Tnoncosingular modules is closed under direct sums if and only if R is a primary decomposable ring. Finitely generated T -noncosingular modules over commutative rings are shown to be precisely those having zero Jacobson radical. We also show that for a simple module S , E(S) ⊕ S is T -noncosingular if and only if S is injective. Connections of T -noncosingular modules to their endomorphism rings are investigated.

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“…, and hence R R is not Tnoncosingular by [20,Corollary 2.7]. On the other hand, we have Z(R R ) = 0 by [4,Corollary 4.3].…”

Section: Example 23 (1) Let R Be a Right V -Ring That Is Not Semisimmentioning

confidence: 87%

“…Proof (i) ⇒ (ii) By [20,Proposition 2.3], N and N ⊕ S i are T -noncosingular modules for all i ∈ I .…”

Section: Ii) If R Is a Ring With Jac(r) = 0 Then Every Free R -Modulmentioning

confidence: 97%

Section: Proposition 41 Let M Be a Quasi-discrete Module Withmentioning

confidence: 99%

Section: Some Properties Of T -Noncosingular Modulesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Some results on T-noncosingular modules

Tribak¹

2014

Turk J Math

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A Generalization of Lifting Modules

Kalati

2015

Ukr Math J

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We introduce the notion of I -lifting modules as a proper generalization of the notion of lifting modules and present some properties of this class of modules. It is shown that if M is an I -lifting direct projective module, then S/r is regular and r = JacS, where S is the ring of all R -endomorphisms of M and r = {φ 2 S | Im φ ⌧ M }. Moreover, we prove that if M is a projective I -lifting module, then M is a direct sum of cyclic modules. The connections between I -lifting modules and dual Rickart modules are presented.

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Direct sums of ADS* modules

Tribak

Tütüncü

Ertaş

2015

Bol. Soc. Mat. Mex.

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A module M is called ADS* if for every direct summand N of M and every supplement K of N in M, we have M = N ⊕ K . In this work, we study direct sums of ADS* modules. Many examples are provided to show that this notion is not inherited by direct sums. It is shown that if a module M has a decomposition M = A ⊕ B which complements direct summands such that A and B are mutually projective, then M is ADS*. The class of rings R, for which all direct sums of ADS* R-modules are ADS*, is shown to be exactly that of the right V-rings. We characterize the class of right perfect rings R for which R ⊕ S is ADS* for every simple R-module S as that of the semisimple rings.

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On 𝒯-Noncosingular Modules

Cited by 13 publications

References 8 publications

Some results on T-noncosingular modules

Some results on T-noncosingular modules

A Generalization of Lifting Modules

Direct sums of ADS* modules

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