½ Cancellation Modules and Homogeneous Idealization II

Ali, Majid

doi:10.1080/00927870802160826

Cited by 13 publications

(10 citation statements)

References 11 publications

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“…Let r 1 ∈ R 1 and m 1 ∈ M 1 such that 0 = r 1 m 1 ∈ N 1 so that (0, 0) = (r 1 , f (r 1 ))(m 1 , ϕ(m 1 )) = (r 1 m 1 , ϕ(r 1 m 1 )) ∈ N 1 ⋊ ⋉ ϕ JM 2 . By assumption, either (s, f (s) + j)(r 1 , f (r 1 )) ∈ (N 1 ⋊ ⋉ ϕ JM 2 : R1⋊ ⋉ f J M 1 ⋊ ⋉ ϕ JM 2 ) or (s, f (s)+j)(m 1 , ϕ(m 1 )) ∈ N 1 ⋊ ⋉ ϕ JM 2 and so N 1 is S-prime in M 1 as in the proof of (1). Now, we use the contrapositive to prove the other part.…”

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

confidence: 93%

“…For a less trivial example, let M be a nonzero local multiplication R-module with the unique maximal submodule K such that (K : R M )K = 0. If we consider S = {1 R }, then every proper submodule of M is weakly S-prime, [1]. Hence, there is a weakly S-prime submodule in M that is not S-prime.…”

Section: Characterizations Of Weakly S-prime Submodulesmentioning

confidence: 99%

“…As usual, Z, Z n and Q denotes the ring of integers, the ring of integers modulo n and the field of rational numbers, respectively. For more details and terminology, one may refer to [1], [2], [8], [14], [15].…”

Section: Introductionmentioning

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: (Weakly) S-prime Submodules Of Amalgamation Modulesmentioning

confidence: 93%

Section: Characterizations Of Weakly S-prime Submodulesmentioning

confidence: 99%

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On weakly S-prime submodules

Khashan¹,

Çeli̇kel²

2021

Preprint

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“…Hasil pertama, terkait gelanggang B́zout dan idealisasinya, diambil dari Cheniour (2012) yang dikutip oleh . Hasil kedua, terkait modul B́zout dan idealisasinya, diambil dari Misri dkk (2016) dan Ali (2007). Hasil kedua ini merupakan perbaikan atas hasil penelitian .…”

Section: Materi Sifat Modul Bezout Perkalian Yang Setiaunclassified

KAJIAN MODUL P-Bézout DAN IDEALISASINYA” UNTUK BUKU AJAR TEORI GELANGGANG BERBASIS RISET

Misri¹

2016

Eduma : Mathematics Education Learning and Teaching

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This paper discusses the properties of P-Bézout modules via idealization. P-Bézout Module here are only Bézout modules. Study of these properties could views in the side of adequacy as a teaching materials on Ring Theory and forming the character of students on other side. Patterns in material transfering process is directly proportional impacted to the student's positive character growth, especially critical and creative thingking as a starting point of teacher's character building which is independent and prestigious since it is able to provide the students wants such as material and personality Abstrak Tulisan ini membahas kajian sifat-sifat modul P-Bézout melalui idealisasi. Modul P-Bézout pada tulisan ini hanya berupa modul Bézout. Kajian sifat-sifat tersebut dilihat dari sisi kecukupan sebagai bahan ajar perkuliahan Teori Gelanggang dan dari sisi pembentukan karakter mahasiswa. Pola penyampaian dalam proses penguasan materi berbanding lurus dengan pertumbuhan karakter positif mahasiswa khususnya berupa sifat kritis dan kreatif sebagai titik tolak terbentuknya pribadi guru yang mandiri dan berwibawa karena mampu memberikan kebutuhan siswa dari sisi materi dan kepribadian.

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“…The class of prime submodules of modules was introduced and studied in 1992 as a generalization of the class of prime ideals of rings. Then, many generalizations of prime submodules were studied such as primary, classical prime, classical primary and classical quasi primary submodules, see [1,8,16,4] and [7].…”

Section: Introductionmentioning

confidence: 99%

Some Remarks on the Classical Prime Spectrum of Modules

Abbasi

Naderi

2021

FU Math Inform

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Let R be a commutative ring with identity and let M be an R-module. A proper submodule P of M is called a classical prime submodule if abm ∈ P, for a,b ∈ R, and m ∈ M, implies that am ∈ P or bm ∈ P. The classical prime spectrum of M, Cl.Spec(M), is deﬁned to be the set of all classical prime submodules of M. We say M is classical primefule if M = 0, or the map ψ from Cl.Spec(M) to Spec(R/Ann(M)), deﬁned by ψ(P) = (P : M)/Ann(M) for all P ∈ Cl.Spec(M), is surjective. In this paper, we study classical primeful modules as a generalisation of primeful modules. Also we investigate some properties of a topology that is deﬁned on Cl.Spec(M), named the Zariski topology.

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½ Cancellation Modules and Homogeneous Idealization II

Cited by 13 publications

References 11 publications

On weakly S-prime submodules

On weakly S-prime submodules

KAJIAN MODUL P-Bézout DAN IDEALISASINYA” UNTUK BUKU AJAR TEORI GELANGGANG BERBASIS RISET

Some Remarks on the Classical Prime Spectrum of Modules

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