PRIMENESS IN MODULE CATEGORY

2010 ◽  
Vol 03 (01) ◽  
pp. 145-154 ◽  
Author(s):  
Nguyen Van Sanh ◽  
Nguyen Anh Vu ◽  
K. F. U. Ahmed ◽  
S. Asawasamrit ◽  
Le Phuong Thao
Keyword(s):  
Module Category ◽  
Prime Ideals ◽  
Prime Submodules ◽  
Associative Rings

We introduce the notion of prime submodules of a given right R-module and describe all properties of them as a generalization of prime ideals in associative rings.

Almost prime submodules

2018 ◽  
Vol 13 (01) ◽  
pp. 2050019
Author(s):  
Le Phuong Thao
Keyword(s):  
Prime Ideals ◽  
Prime Submodules ◽  
Associative Rings ◽  
Almost Prime ◽  
Weakly Prime Ideals

In this paper, we introduce the notion of almost prime submodules of a given right [Formula: see text]-module and study some properties of them as a generalization of weakly prime ideals in associative rings. We will also investigate modules in which every fully invariant submodule is almost prime.

Zariski-like topologies for lattices with applications to modules over associative rings

2019 ◽  
Vol 18 (07) ◽  
pp. 1950131
Author(s):  
Jawad Abuhlail ◽  
Hamza Hroub
Keyword(s):  
Complete Lattice ◽  
Associative Ring ◽  
Sufficient Conditions ◽  
Proper Class ◽  
Topological Properties ◽  
Left Module ◽  
Separation Axioms ◽  
Prime Submodules ◽  
Algebraic Properties ◽  
Associative Rings

We study Zariski-like topologies on a proper class [Formula: see text] of a complete lattice [Formula: see text]. We consider [Formula: see text] with the so-called classical Zariski topology [Formula: see text] and study its topological properties (e.g. the separation axioms, the connectedness, the compactness) and provide sufficient conditions for it to be spectral. We say that [Formula: see text] is [Formula: see text]-top if [Formula: see text] is a topology. We study the interplay between the algebraic properties of an [Formula: see text]-top complete lattice [Formula: see text] and the topological properties of [Formula: see text] Our results are applied to several spectra which are proper classes of [Formula: see text] where [Formula: see text] is a nonzero left module over an arbitrary associative ring [Formula: see text] (e.g. the spectra of prime, coprime, fully prime submodules) of [Formula: see text] as well as to several spectra of the dual complete lattice [Formula: see text] (e.g. the spectra of first, second and fully coprime submodules of [Formula: see text]).

scholarly journals A note on the countable union of prime submodules

2001 ◽  
Vol 27 (10) ◽  
pp. 641-643 ◽  
Author(s):  
M. R. Pournaki ◽  
M. Tousi
Keyword(s):  
Noetherian Ring ◽  
Countable Family ◽  
Prime Ideals ◽  
Countable Union ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Prime Submodules

LetMbe a finitely-generated module over a Noetherian ringR. Suppose𝔞is an ideal ofRand letN=𝔞Mand𝔟=Ann(M/N). If𝔟⫅J(R),Mis complete with respect to the𝔟-adic topology,{Pi}i≥1is a countable family of prime submodules ofM, andx∈M, thenx+N⫅∪i≥1Piimplies thatx+N⫅Pjfor somei≥1. This extends a theorem of Sharp and Vámos concerning prime ideals to prime submodules.

scholarly journals Zariski topology on the spectrum of graded pseudo prime submodules

2021 ◽  
Vol 39 (3) ◽  
pp. 17-26
Author(s):  
Rashid Abu-Dawwas
Keyword(s):  
Topological Space ◽  
Commutative Rings ◽  
Prime Ideals ◽  
Zariski Topology ◽  
Graded Modules ◽  
Prime Submodules ◽  
Algebraic Properties

In this article, we introduce the concept of graded pseudo prime submodules of graded modules that is a generalization of the graded prime ideals over commutative rings. We study the Zariski topology on the graded spectrum of graded pseudo prime submodules. We clarify the relation between the properties of this topological space and the algebraic properties of the graded modules under consideration.

scholarly journals On alpha-prime and Weakly alpha-prime Submodules

2018 ◽  
Vol 11 (3) ◽  
pp. 730-739 ◽  
Author(s):  
Thawatchai Khumprapussorn
Keyword(s):  
Prime Ideals ◽  
Basic Properties ◽  
Prime Submodules

We have introduced the notion of $\alpha$-prime and weakly $\alpha$-prime submodules as a generalization of prime submodules. Some basic properties of $\alpha$-prime and weakly $\alpha$-prime submodules are the extension of prime submodules. Finally, after introducing the notion of $\alpha$-prime submodules, we also define and study the concept of $\alpha$-prime ideals in a ring.

scholarly journals On radical formula in modules over noncommutative rings

Filomat ◽  
2020 ◽  
Vol 34 (2) ◽  
pp. 443-449
Author(s):  
Ortac Öneş
Keyword(s):  
Prime Ideals ◽  
Direct Connection ◽  
Noncommutative Rings ◽  
Nilpotent Elements ◽  
Prime Submodules ◽  
Prime Submodule ◽  
Strongly Nilpotent

This paper examines the radical formula in noncommutative case and for this purpose, a generalization of prime submodule is defined. It is proved that there is a direct connection between onesided prime ideals and one-sided prime submodules. Moreover the connections between the intersection of all one-sided prime submodules and strongly nilpotent elements of a module are studied.

Sifat-sifat Submodul Prima dan Submodul Prima Lemah

2020 ◽  
Vol 2 (2) ◽  
pp. 183
Author(s):  
Hisyam Ihsan ◽  
Muhammad Abdy ◽  
Samsu Alam B
Keyword(s):  
Prime Ideal ◽  
Literature Study ◽  
Prime Submodules ◽  
Unitary Module ◽  
Prime Submodule ◽  
Definition Of ◽  
The Relationship

Penelitian ini merupakan penelitian kajian pustaka yang bertujuan untuk mengkaji sifat-sifat submodul prima dan submodul prima lemah serta hubungan antara keduanya. Kajian dimulai dari definisi submodul prima dan submodul prima lemah, selanjutnya dikaji mengenai sifat-sifat dari keduanya. Pada penelitian ini, semua ring yang diberikan adalah ring komutatif dengan unsur kesatuan dan modul yang diberikan adalah modul uniter. Sebagai hasil dari penelitian ini diperoleh beberapa pernyataan yang ekuivalen, misalkan  suatu -modul ,  submodul sejati di  dan ideal di , maka ketiga pernyataan berikut ekuivalen, (1)  merupakan submodul prima, (2) Setiap submodul tak nol dari   -modul memiliki annihilator yang sama, (3) Untuk setiap submodul  di , subring  di , jika berlaku  maka  atau . Di lain hal, pada submodul prima lemah jika diberikan  suatu -modul,  submodul sejati di , maka pernyataan berikut ekuivalen, yaitu (1) Submodul  merupakan submodul prima lemah, (2) Untuk setiap , jika  maka . Selain itu, didapatkan pula hubungan antara keduanya, yaitu setiap submodul prima merupakan submodul prima lemah.Kata Kunci: Submodul Prima, Submodul Prima Lemah, Ideal Prima. This research is literature study that aims to examine the properties of prime submodules and weakly prime submodules and the relationship between  both of them. The study starts from the definition of prime submodules and weakly prime submodules, then reviewed about the properties both of them. Throughout this paper all rings are commutative with identity and all modules are unitary. As the result of this research, obtained several equivalent statements, let  be a -module,  be a proper submodule of  and  ideal of , then the following three statetments are equivalent, (1)  is a prime submodule, (2) Every nonzero submodule of   -module has the same annihilator, (3) For any submodule  of , subring  of , if  then  or . In other case, for weakly prime submodules, if given  is a unitary -module,  be a proper submodule of , then the following statements are equivalent, (1)  is a weakly prime submodule, (2) For any , if  then . In addition, also found the relationship between both of them, i.e. any prime submodule is weakly prime submodule.Keywords: Prime Submodules, Weakly Prime Submdules, Prime Ideal.

Small Prime Modules and Small Prime Submodules

2012 ◽  
Vol 15 (4) ◽  
pp. 191-199
Author(s):  
Layla S. Mahmood ◽  
Keyword(s):  
Prime Submodules

N-ideals of commutative rings

Filomat ◽  
2017 ◽  
Vol 31 (10) ◽  
pp. 2933-2941 ◽  
Author(s):  
Unsal Tekir ◽  
Suat Koc ◽  
Kursat Oral
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Proper Ideal ◽  
Prime Ideals ◽  
Nonzero Identity

In this paper, we present a new classes of ideals: called n-ideal. Let R be a commutative ring with nonzero identity. We define a proper ideal I of R as an n-ideal if whenever ab ? I with a ? ?0, then b ? I for every a,b ? R. We investigate some properties of n-ideals analogous with prime ideals. Also, we give many examples with regard to n-ideals.

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