On Fully Idempotent Modules

Tütüncü, Derya Keskı̇n; Ertaş, Nil Orhan; Tribak, Rachid; Smith, Patrick F.

doi:10.1080/00927872.2010.489916

Cited by 4 publications

(8 citation statements)

References 10 publications

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“…+ f n (x n ) . In [2] , M is called fully idempotent if every submodule of M is idempotent in M . Now ,If M is a module over a commutative ringwith1.…”

Section: ) Let a Be Any Submodule Of M And B Be A Relative Complmentmentioning

confidence: 99%

“…Thus A is an idempotent submodule of M . Recall that an Rmodule M is called fully idempotent if every submodule of M is idempotent , [2] . Now , we give a characterization for fully idempotent modules .…”

Section: ) Let a Be Any Submodule Of M And B Be A Relative Complmentmentioning

confidence: 99%

“…Let R be an associative ring withidentity , and let M be a (left) unitary module . Following [2] a submoduleA of a module M is called idempotent submodule of M provided N = Hom (M,A) A = . That is A is an idempotent submoduleof M if for each xN , there exist a positive integer k , homomorphismsf i : M A (1ik) such that x = f 1 (x 1 ) + ….…”

Section: Introductionmentioning

confidence: 99%

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On Strongly F – Regular Modules and Strongly Pure Intersection Property

Bahrani¹

2014

Baghdad Sci.J

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A submoduleA of amodule M is said to be strongly pure , if for each finite subset {a i } in A , (equivalently, for each a A) there exists ahomomorphism f : M A such that f(a i ) = a i , i(f(a)=a). A module M is said to be strongly F-regular if each submodule of M is strongly pure . The main purpose of this paper is to develop the properties of strongly F-regular modules and study modules with the property that the intersection of any two strongly pure submodules is strongly pure .

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“…+ f n (x n ) . In [2] , M is called fully idempotent if every submodule of M is idempotent in M . Now ,If M is a module over a commutative ringwith1.…”

Section: ) Let a Be Any Submodule Of M And B Be A Relative Complmentmentioning

confidence: 99%

Section: ) Let a Be Any Submodule Of M And B Be A Relative Complmentmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On Strongly F – Regular Modules and Strongly Pure Intersection Property

Bahrani¹

2014

Baghdad Sci.J

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“…10 and 6.7] and in the characterization of endomorphism rings of quasi-injective envelopes of polyform modules [5, 5.19]; see also, [9,Theorem 2.6], and [24,Section 2]. In [21], it is shown that the commutative rings over which every module is fully idempotent are exactly the semisimple rings.…”

Section: Introductionmentioning

confidence: 99%

“…Following [12], an R-module M is called retractable if Hom R (M, N ) = 0 for all nonzero submodules N of M . Semisimple modules and fully idempotent modules [21] are clearly retractable, and more generally self-projective modules with zero radical and essentially compressible modules are known to enjoy this property; see [5, 3.4] and [19,Theorem 3.1]. Retractable modules have appeared in different situations.…”

Section: Introductionmentioning

confidence: 99%

On rings whose modules have nonzero homomorphisms to nonzero submodules

Tolooei¹,

Vedadi²

2013

Publicacions Matemàtiques

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We carry out a study of rings R for which Hom R (M, N ) = 0 for all nonzero N ≤ M R . Such rings are called retractable. For a retractable ring, Artinian condition and having Krull dimension are equivalent. Furthermore, a right Artinian ring in which prime ideals commute is precisely a right Noetherian retractable ring. Retractable rings are characterized in several ways. They form a class of rings that properly lies between the class of pseudo-Frobenius rings, and the class of max divisible rings for which the converse of Schur's lemma holds. For several types of rings, including commutative rings, retractability is equivalent to semi-Artinian condition. We show that a Köthe ring R is an Artinian principal ideal ring if and only if it is a certain retractable ring, and determine when R is retractable.

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A characterization of modules embedding in products of primes and enveloping condition for their class

Dehghani

Vedadi

2015

J. Algebra Appl.

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Modules (cogenerated by nonzero their submodules) having nonzero square homomorphism in nonzero submodules are said (prime) weakly compressible (wc). Such modules are semiprime (i.e. they are cogenerated by their essential submodules). For many rings R, including commutative rings, it is proved that wc modules are isomorphic submodules of products of prime modules, and the converse holds precisely when the class of wc modules is enveloping for mod-R or equivalently, every semiprime module is wc. Semi-Artinian rings R have the latter property and the converse is true when R is strongly regular. Duo Noetherian rings over which wc modules form an enveloping class are shown to have a finite number maximal ideals. If R is Morita equivalent to a Dedekind domain, then the class of wc R-modules is enveloping if and only if R is simple Artinian or J (R) ≠ 0.

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On Fully Idempotent Modules

Cited by 4 publications

References 10 publications

On Strongly F – Regular Modules and Strongly Pure Intersection Property

On Strongly F – Regular Modules and Strongly Pure Intersection Property

On rings whose modules have nonzero homomorphisms to nonzero submodules

A characterization of modules embedding in products of primes and enveloping condition for their class

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