Small Prime Modules and Small Prime Submodules

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

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.

Prime uniserial modules and rings

2019 ◽  
Vol 18 (02) ◽  
pp. 1950035 ◽  
Author(s):  
M. Behboodi ◽  
Z. Fazelpour
Keyword(s):  
Commutative Rings ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Direct Sum ◽  
Dedekind Domains ◽  
Prime Submodules ◽  
Uniserial Modules ◽  
Structure Formula

We define prime uniserial modules as a generalization of uniserial modules. We say that an [Formula: see text]-module [Formula: see text] is prime uniserial ([Formula: see text]-uniserial) if its prime submodules are linearly ordered by inclusion, and we say that [Formula: see text] is prime serial ([Formula: see text]-serial) if it is a direct sum of [Formula: see text]-uniserial modules. The goal of this paper is to study [Formula: see text]-serial modules over commutative rings. First, we study the structure [Formula: see text]-serial modules over almost perfect domains and then we determine the structure of [Formula: see text]-serial modules over Dedekind domains. Moreover, we discuss the following natural questions: “Which rings have the property that every module is [Formula: see text]-serial?” and “Which rings have the property that every finitely generated module is [Formula: see text]-serial?”.

scholarly journals Projective modules and prime submodules

2006 ◽  
Vol 56 (2) ◽  
pp. 601-611 ◽  
Author(s):  
Mustafa Alkan ◽  
Yücel Tiraş
Keyword(s):  
Projective Modules ◽  
Prime Submodules

scholarly journals On a generalization of prime submodules of a module over a commutative ring

2017 ◽  
Vol 37 (1) ◽  
pp. 153-168
Author(s):  
Hosein Fazaeli Moghimi ◽  
Batool Zarei Jalal Abadi
Keyword(s):  
Commutative Ring ◽  
Dedekind Domain ◽  
Finitely Generated ◽  
Multiplication Module ◽  
Maximal Ideals ◽  
Prime Submodules ◽  
Prime Submodule

‎Let $R$ be a commutative ring with identity‎, ‎and $n\geq 1$ an integer‎. ‎A proper submodule $N$ of an $R$-module $M$ is called‎ ‎an $n$-prime submodule if whenever $a_1 \cdots a_{n+1}m\in N$ for some non-units $a_1‎, ‎\ldots‎ , ‎a_{n+1}\in R$ and $m\in M$‎, ‎then $m\in N$ or there are $n$ of the $a_i$'s whose product is in $(N:M)$‎. ‎In this paper‎, ‎we study $n$-prime submodules as a generalization of prime submodules‎. ‎Among other results‎, ‎it is shown that if $M$ is a finitely generated faithful multiplication module over a Dedekind domain $R$‎, ‎then every $n$-prime submodule of $M$ has the form $m_1\cdots m_t M$ for some maximal ideals $m_1,\ldots,m_t$ of $R$ with $1\leq t\leq n$‎.

Strongly 0-dimensional Modules

2014 ◽  
Vol 57 (1) ◽  
pp. 159-165 ◽  
Author(s):  
Kürşat Hakan Oral ◽  
Neslihan Ayşen Özkirişci ◽  
Ünsal Tekir
Keyword(s):  
Multiplication Module ◽  
Finite Intersection ◽  
Von Neumann ◽  
Prime Submodules ◽  
Von Neumann Regular ◽  
Prime Submodule ◽  
Dimensional Module

AbstractIn a multiplication module, prime submodules have the following property: if a prime submodule contains a finite intersection of submodules, then one of the submodules is contained in the prime submodule. In this paper, we generalize this property to infinite intersection of submodules and call such prime submodules strongly prime submodules. A multiplication module in which every prime submodule is strongly prime will be called a strongly 0-dimensional module. It is also an extension of strongly 0-dimensional rings. After this we investigate properties of strongly 0-dimensional modules and give relations of von Neumann regular modules, Q-modules and strongly 0-dimensional modules

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

2013 ◽  
Vol 50 (6) ◽  
pp. 1183-1198
Author(s):  
Ashkan Nikseresht ◽  
Abdulrasool Azizi
Keyword(s):  
Prime Submodules

scholarly journals On Graded Quasi-Prime Submodules

2015 ◽  
Vol 55 (2) ◽  
pp. 259-266 ◽  
Author(s):  
KHALDOUN AL-ZOUBI ◽  
RASHID ABU-DAWWAS
Keyword(s):  
Prime Submodules

scholarly journals Prime Submodules, the Descent Invariant, and Modules of Finite Length

1997 ◽  
Vol 189 (2) ◽  
pp. 273-293 ◽  
Author(s):  
Agustı́n Marcelo ◽  
J. Muñoz Masqué
Keyword(s):  
Finite Length ◽  
Prime Submodules

A GENERALIZATION OF BAER'S LOWER NILRADICAL FOR MODULES

2007 ◽  
Vol 06 (02) ◽  
pp. 337-353 ◽  
Author(s):  
MAHMOOD BEHBOODI
Keyword(s):  
Prime Ring ◽  
Artinian Ring ◽  
Prime Radical ◽  
Nilpotent Elements ◽  
Noetherian Domain ◽  
Prime Submodules ◽  
Strongly Nilpotent ◽  
Commutative Domain ◽  
Torsion Modules

Let M be a left R-module. A proper submodule P of M is called classical prime if for all ideals [Formula: see text] and for all submodules N ⊆ M, [Formula: see text] implies that [Formula: see text] or [Formula: see text]. We generalize the Baer–McCoy radical (or classical prime radical) for a module [denoted by cl.rad R(M)] and Baer's lower nilradical for a module [denoted by Nil *(RM)]. For a module RM, cl.rad R(M) is defined to be the intersection of all classical prime submodules of M and Nil *(RM) is defined to be the set of all strongly nilpotent elements of M (defined later). It is shown that, for any projective R-module M, cl.rad R(M) = Nil *(RM) and, for any module M over a left Artinian ring R, cl.rad R(M) = Nil *(RM) = Rad (M) = Jac (R)M. In particular, if R is a commutative Noetherian domain with dim (R) ≤ 1, then for any module M, we have cl.rad R(M) = Nil *(RM). We show that over a left bounded prime left Goldie ring, the study of Baer–McCoy radicals of general modules reduces to that of torsion modules. Moreover, over an FBN prime ring R with dim (R) ≤ 1 (or over a commutative domain R with dim (R) ≤ 1), every semiprime submodule of any module is an intersection of classical prime submodules.

ON GRADED S-PRIME SUBMODULES OF GRADED MODULES OVER GRADED COMMUTATIVE RINGS

2021 ◽  
Vol 10 (11) ◽  
pp. 3479-3489
Author(s):  
K. Al-Zoubi ◽  
M. Al-Azaizeh
Keyword(s):  
Abelian Group ◽  
Commutative Ring ◽  
Closed Subset ◽  
Commutative Rings ◽  
Graded Modules ◽  
Prime Submodules ◽  
Prime Submodule

Let $G$ be an abelian group with identity $e$. Let $R$ be a $G$-graded commutative ring with identity, $M$ a graded $R$-module and $S\subseteq h(R)$ a multiplicatively closed subset of $R$. In this paper, we introduce the concept of graded $S$-prime submodules of graded modules over graded commutative rings. We investigate some properties of this class of graded submodules and their homogeneous components. Let $N$ be a graded submodule of $M$ such that $(N:_{R}M)\cap S=\emptyset $. We say that $N$ is \textit{a graded }$S$\textit{-prime submodule of }$M$ if there exists $s_{g}\in S$ and whenever $a_{h}m_{i}\in N,$ then either $s_{g}a_{h}\in (N:_{R}M)$ or $s_{g}m_{i}\in N$ for each $a_{h}\in h(R) $ and $m_{i}\in h(M).$

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