On Supplement Submodules

Nebiyev, Celil; Pancar, Ali

doi:10.1007/s11253-013-0842-2

Cited by 11 publications

(17 citation statements)

References 5 publications

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“…It was shown in [5,Theorem 2.1] that if F is a supplement of a submodule K in a module M , then it is possible to define a bijective map between maximal submodules of F and maximal submodules of M which contain K. In the next result, we use this fact to characterize supplement submodules in a coatomic module.…”

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confidence: 93%

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Supplements in Coatomic Modules Having the Complete Max-Property

Cisse¹,

Ngom²,

Sow³

et al. 2018

International Electronic Journal of Algebra

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Let R be a ring with identity. A right R-module M has the complete max-property if the maximal submodules of M are completely coindependent (i.e., every maximal submodule of M does not contain the intersection of the other maximal submodules of M ). A right R-module is said to be a good module provided every proper submodule of M containing Rad(M ) is an intersection of maximal submodules of M . We obtain a new characterization of good modules. Also, we study good modules which have the complete maxproperty. The second part of this paper is devoted to investigate supplements in a coatomic module which has the complete max-property. Mathematics Subject Classification (2010): 16D10, 16D99 Let R be a unitary ring and M a right R-module. A submodule N of M is called small in M (written N M ) if for every proper submodule L of M , N + L = M . A submodule L of M is called coclosed in M if L/K is not small in M/K for any proper submodule K of L. We denote by Rad(M ) the radical of M . A module M is called coatomic if every proper submodule of M is contained in a maximal submodule, that is, Rad(M/N ) = 0 for every proper submodule N ≤ M . Let L be a submodule of M . A submodule K of M is called a supplement of L in M if K is minimal with respect to the property M = L + K; equivalently, M = L + Kand K ∩ L K. A submodule P of M is called a supplement submodule if P is a supplement of some submodule of M . The module M is called supplemented if every submodule of M has a supplement in M . A module M is called semilocal if M/ Rad(M ) is semisimple. A module M is called cosemisimple (or a V-module) if every simple R-module is M -injective, or equivalently, every proper submodule of M is an intersection of maximal submodules (see [7, 23.1]). A module M is called a SUPPLEMENTS IN COATOMIC MODULES 19 good module if M/ Rad(M ) is a cosemisimple module (see [7, 23.3]). A non-empty family of submodules N i (i ∈ I) of a module M is called coindependent if, for any j ∈ I and any finite subset J of I \ {j}, N j + i∈J N i = M . The family N i (i ∈ I) is called completely coindependent if, for every j ∈ I, N j + i =j N i = M (see [4, p. 8]). Following [6, p. 74], a module M is said to have the complete max-property if the maximal submodules of M form a completely coindependent set of submodules of M . In this paper, we adopt the convention that the intersection of an empty set of submodules of a module M is M itself.In Section 2, we provide some new characterizations of good modules (Theorem 2.3). Also, we investigate the interplay between the complete max-property and each one of the properties coatomic and good.The investigations in Section 3 focus on supplements in a coatomic module which has the complete max-property. After characterizing them, we show that for a coatomic module M , if M has the complete max-property, then any supplement submodule in M has also the complete max-property. In addition, we prove that if M is a coatomic module which has the complete max-property and F is a supple-Throughout this paper, R will denote an ...

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confidence: 93%

“…Without loss of generality we can assume that M 1 ⊆ N . Since M 2 is a supplement of M 1 , the maximal submodules of M 2 are {N i ∩ M 2 | i ∈ I} where {N i | i ∈ I} are the maximal submodules of M containing M 1 by[5, Theorem 2.1]. So N = N i0 for some i 0 ∈ I.…”

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confidence: 99%

Supplements in Coatomic Modules Having the Complete Max-Property

Cisse¹,

Ngom²,

Sow³

et al. 2018

International Electronic Journal of Algebra

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“…More informations about (amply) supplemented lattices are in [1,2,5,9]. More results about (amply) supplemented modules are in [8,12]. The definition of β * relation on lattices and some properties of this relation are in [10].…”

Section: Introductionmentioning

confidence: 99%

Essential supplemented lattices

Ökten¹,

Pekin²

2020

MMN

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“…More results about (amply) supplemented modules are in [5,9]. Some important properties of supplement submodules are in [8]. The definition of generalized supplemented lattices and some properties of them are in [3].…”

Section: Introductionmentioning

confidence: 99%

“…Some relation between lying above and (weak) supplement elements also studied. Some properties of supplement submodules in modules which given in [8] are generalized to lattices. Let a be a supplement of b in a lattice L. If a=0 has at least one maximal .¤ a/ element, then it is possible to define a bijective map between the maximal elements .¤ a/ of a=0 and the maximal elements .¤ 1/ of 1=b.…”

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confidence: 99%

On supplement elements in lattices

Nebiyev¹

2019

MMN

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In this work, some properties of supplement elements in lattices are investigated. Some relation between lying above and (weak) supplement elements also studied. Some properties of supplement submodules in modules which given in [8] are generalized to lattices. Let a be a supplement of b in a lattice L. If a=0 has at least one maximal .¤ a/ element, then it is possible to define a bijective map between the maximal elements .¤ a/ of a=0 and the maximal elements .¤ 1/ of 1=b. Let a be a supplement element in a lattice L. If L is amply supplemented, then a=0 is also amply supplemented. If L is weakly supplemented, then a=0 is also weakly supplemented.

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On Supplement Submodules

Cited by 11 publications

References 5 publications

Supplements in Coatomic Modules Having the Complete Max-Property

Supplements in Coatomic Modules Having the Complete Max-Property

Essential supplemented lattices

On supplement elements in lattices

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