• Title/Summary/Keyword: Prime submodule

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Weakly Classical Prime Submodules

  • Mostafanasab, Hojjat;Tekir, Unsal;Oral, Kursat Hakan
    • Kyungpook Mathematical Journal
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    • v.56 no.4
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    • pp.1085-1101
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    • 2016
  • In this paper, all rings are commutative with nonzero identity. Let M be an R-module. A proper submodule N of M is called a classical prime submodule, if for each $m{\in}M$ and elements a, $b{\in}R$, $abm{\in}N$ implies that $am{\in}N$ or $bm{\in}N$. We introduce the concept of "weakly classical prime submodules" and we will show that this class of submodules enjoys many properties of weakly 2-absorbing ideals of commutative rings. A proper submodule N of M is a weakly classical prime submodule if whenever $a,b{\in}R$ and $m{\in}M$ with $0{\neq}abm{\in}N$, then $am{\in}N$ or $bm{\in}N$.

On Weakly Prime and Weakly 2-absorbing Modules over Noncommutative Rings

  • Groenewald, Nico J.
    • Kyungpook Mathematical Journal
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    • v.61 no.1
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    • pp.33-48
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    • 2021
  • Most of the research on weakly prime and weakly 2-absorbing modules is for modules over commutative rings. Only scatterd results about these notions with regard to non-commutative rings are available. The motivation of this paper is to show that many results for the commutative case also hold in the non-commutative case. Let R be a non-commutative ring with identity. We define the notions of a weakly prime and a weakly 2-absorbing submodules of R and show that in the case that R commutative, the definition of a weakly 2-absorbing submodule coincides with the original definition of A. Darani and F. Soheilnia. We give an example to show that in general these two notions are different. The notion of a weakly m-system is introduced and the weakly prime radical is characterized interms of weakly m-systems. Many properties of weakly prime submodules and weakly 2-absorbing submodules are proved which are similar to the results for commutative rings. Amongst these results we show that for a proper submodule Ni of an Ri-module Mi, for i = 1, 2, if N1 × N2 is a weakly 2-absorbing submodule of M1 × M2, then Ni is a weakly 2-absorbing submodule of Mi for i = 1, 2

PRIME BASES OF WEAKLY PRIME SUBMODULES AND THE WEAK RADICAL OF SUBMODULES

  • Nikseresht, Ashkan;Azizi, Abdulrasool
    • Journal of the Korean Mathematical Society
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    • v.50 no.6
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    • pp.1183-1198
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    • 2013
  • We will introduce and study the notion of prime bases for weakly prime submodules and utilize them to derive some formulas on the weak radical of submodules of a module. In particular, we will show that every one dimensional integral domain weakly satisfies the radical formula and state some necessary conditions on local integral domains which are semi-compatible or satisfy the radical formula and also on Noetherian rings which weakly satisfy the radical formula.

ON PRIME SUBMODULES OF A FINITELY GENERATED PROJECTIVE MODULE OVER A COMMUTATIVE RING

  • Nekooei, Reza;Pourshafiey, Zahra
    • Communications of the Korean Mathematical Society
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    • v.34 no.3
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    • pp.729-741
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    • 2019
  • In this paper we give a full characterization of prime submodules of a finitely generated projective module M over a commutative ring R with identity. Also we study the existence of primary decomposition of a submodule of a finitely generated projective module and characterize the minimal primary decomposition of this submodule. Finally, we characterize the radical of an arbitrary submodule of a finitely generated projective module M and study submodules of M which satisfy the radical formula.

THE NILPOTENCY OF THE PRIME RADICAL OF A GOLDIE MODULE

  • John A., Beachy;Mauricio, Medina-Barcenas
    • Bulletin of the Korean Mathematical Society
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    • v.60 no.1
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    • pp.185-201
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    • 2023
  • With the notion of prime submodule defined by F. Raggi et al. we prove that the intersection of all prime submodules of a Goldie module M is a nilpotent submodule provided that M is retractable and M(Λ)-projective for every index set Λ. This extends the well known fact that in a left Goldie ring the prime radical is nilpotent.

ON DISTINGUISHED PRIME SUBMODULES

  • Cho, Yong-Hwan
    • Communications of the Korean Mathematical Society
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    • v.15 no.3
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    • pp.493-498
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    • 2000
  • In this paper we find some properties of distinguished prime submodules of modules and prove theorems about the dimension of modules.

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A NOTE ON MONOFORM MODULES

  • Hajikarimi, Alireza;Naghipour, Ali Reza
    • Bulletin of the Korean Mathematical Society
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    • v.56 no.2
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    • pp.505-514
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    • 2019
  • Let R be a commutative ring with identity and M be a unitary R-module. A submodule N of M is called a dense submodule if $Hom_R(M/N,\;E_R(M))=0$, where $E_R(M)$ is the injective hull of M. The R-module M is said to be monoform if any nonzero submodule of M is a dense submodule. In this paper, among the other results, it is shown that any kind of the following module is monoform. (1) The prime R-module M such that for any nonzero submodule N of M, $Ann_R(M/N){\neq}Ann_R(M)$. (2) Strongly prime R-module. (3) Faithful multiplication module over an integral domain.

ON WEAKLY S-PRIME SUBMODULES

  • Hani A., Khashan;Ece Yetkin, Celikel
    • Bulletin of the Korean Mathematical Society
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    • v.59 no.6
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    • pp.1387-1408
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    • 2022
  • 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 = ∅ 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.

PRIME M-IDEALS, M-PRIME SUBMODULES, M-PRIME RADICAL AND M-BAER'S LOWER NILRADICAL OF MODULES

  • Beachy, John A.;Behboodi, Mahmood;Yazdi, Faezeh
    • Journal of the Korean Mathematical Society
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    • v.50 no.6
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    • pp.1271-1290
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    • 2013
  • Let M be a fixed left R-module. For a left R-module X, we introduce the notion of M-prime (resp. M-semiprime) submodule of X such that in the case M=R, it coincides with prime (resp. semiprime) submodule of X. Other concepts encountered in the general theory are M-$m$-system sets, M-$n$-system sets, M-prime radical and M-Baer's lower nilradical of modules. Relationships between these concepts and basic properties are established. In particular, we identify certain submodules of M, called "primeM-ideals", that play a role analogous to that of prime (two-sided) ideals in the ring R. Using this definition, we show that if M satisfies condition H (defined later) and $Hom_R(M,X){\neq}0$ for all modules X in the category ${\sigma}[M]$, then there is a one-to-one correspondence between isomorphism classes of indecomposable M-injective modules in ${\sigma}[M]$ and prime M-ideals of M. Also, we investigate the prime M-ideals, M-prime submodules and M-prime radical of Artinian modules.

ON NOETHERIAN PSEUDO-PRIME SPECTRUM OF A TOPOLOGICAL LE-MODULE

  • Anjan Kumar Bhuniya;Manas Kumbhakar
    • Communications of the Korean Mathematical Society
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    • v.38 no.1
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    • pp.1-9
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    • 2023
  • An le-module M over a commutative ring R is a complete lattice ordered additive monoid (M, ⩽, +) having the greatest element e together with a module like action of R. This article characterizes the le-modules RM such that the pseudo-prime spectrum XM endowed with the Zariski topology is a Noetherian topological space. If the ring R is Noetherian and the pseudo-prime radical of every submodule elements of RM coincides with its Zariski radical, then XM is a Noetherian topological space. Also we prove that if R is Noetherian and for every submodule element n of M there is an ideal I of R such that V (n) = V (Ie), then the topological space XM is spectral.