Duplication of a module along an ideal

Bouba, El Mehdi; Mahdou, Najib; Tamekkante, Mohammed

doi:10.1007/s10474-017-0775-6

Cited by 9 publications

(8 citation statements)

References 12 publications

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“…are submodules of M 1 ⋊ ⋉ ϕ JM 2 . Section 3 is devoted for studying several conditions under which the submodules N 1 ⋊ ⋉ ϕ JM 2 and N 2 ϕ of M 1 ⋊ ⋉ ϕ JM 2 are (weakly) Sprime submodules, (see Theorems 8,10). Furthermore, we conclude some particular results for the duplication of a module along an ideal (see Corollaries 4-6, 7-9 and Theorem 9).…”

Section: Introductionmentioning

confidence: 85%

“…is a subring of R × R called the the amalgamated duplication of R along J, see [11]. Recently, in [10], the duplication of the R-module M along the ideal J denoted by M ⋊ ⋉ J is defined as…”

Section: (Weakly) S-prime Submodules Of Amalgamation Modulesmentioning

confidence: 99%

“…(m, m ′ ) = (rm, (r +j)m ′ ) for r ∈ R, j ∈ J and (m, m ′ ) ∈ M ⋊ ⋉ J. Many properties and results concerning this kind of modules can be found in [10].…”

Section: (Weakly) S-prime Submodules Of Amalgamation Modulesmentioning

confidence: 99%

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On weakly S-prime submodules

Khashan¹,

Çeli̇kel²

2021

Preprint

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Let R be a commutative ring with a non-zero identity, S be a multiplicatively closed subset of R and M be a unital R-module. In this paper, we define a submodule N of M with (N : R M ) ∩ S = φ to be weakly S-prime if there exists s ∈ S such that whenever a ∈ R and m ∈ M with 0 = am ∈ N , then either sa ∈ (N : R M ) or sm ∈ N . Many properties, examples and characterizations of weakly S-prime submodules are introduced, especially in multiplication modules. Moreover, we investigate the behavior of this structure under module homomorphisms, localizations, quotient modules, cartesian product and idealizations. Finally, we define two kinds of submodules of the amalgamation module along an ideal and investigate conditions under which they are weakly S-prime.

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Section: Introductionmentioning

confidence: 85%

Section: (Weakly) S-prime Submodules Of Amalgamation Modulesmentioning

confidence: 99%

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On weakly S-prime submodules

Khashan¹,

Çeli̇kel²

2021

Preprint

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“…A ⊲⊳ f J := {(a, f (a) + j) | a ∈ A, j ∈ J} called the amalgamation of A with B along J with respect to f . This construction is a generalization of the amalgamated duplication of a ring along an ideal (cf., for instance [6,8,9,10,11,12]).…”

Section: Amalgamation Of S-1-absorbing Primary Submodulesmentioning

confidence: 99%

S-1-absorbing primary submodules

Issoual¹,

Mahdou²,

Ozkirisci³

et al. 2022

Preprint

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In this work, we introduce the notion of S-1-absorbing primary submodule as an extension of 1-absorbing primary submodule. Let S be a multiplicatively closed subset of a ring R and M be an R-module. A submodule N of M with (N : R M )∩S = ∅ is said to be S-1-absorbing primary if whenever abm ∈ N for some non-unit a, b ∈ R and m ∈ M , then either sab ∈ (N : R M ) or sm ∈ M -rad(N ). We examine several properties of this concept and provide some characterizations. In addition, S-1-absorbing primary avoidance theorem and S-1-absorbing primary property for idealization and amalgamation are presented.

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“…Let A be a ring, I an ideal of A, and M an A-module. The authors of [6] If M = A, then the duplication of the A-module A along the ideal I coincides with the amalgamated duplication of the ring A along the ideal I. In their article, they studied some basic properties of the duplication of an A-module M along an ideal I.…”

Section: Introductionmentioning

confidence: 99%