On the Dual Notion of Prime Submodules

2012 ◽  
Vol 19 (spec01) ◽  
pp. 1109-1116 ◽  
Author(s):  
H. Ansari-Toroghy ◽  
F. Farshadifar
Keyword(s):  
Commutative Ring ◽  
Prime Submodules ◽  
Dual Notion

Let R be a commutative ring and M an R-module. In this paper, we study the dual notion of prime submodules (that is, second submodules of M) and investigate the conditions under which the number of maximal second submodules of M is finite. Furthermore, we introduce the concept of coisolated submodules of M and obtain some related characterizations.

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 $1\leq t\leq n$‎.

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 $m_{i}\in h(M).$

scholarly journals On graded Ie-prime submodules of graded modules over graded commutative rings

2021 ◽  
Vol 110 (124) ◽  
pp. 47-55
Author(s):  
Shatha Alghueiri ◽  
Khaldoun Al-Zoubi
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Graded Modules ◽  
Prime Submodules ◽  
Prime Submodule

Let G be a group with identity e. Let R be a G-graded commutative ring with identity and M a graded R-module. We introduce the concept of graded Ie-prime submodule as a generalization of a graded prime submodule for I =?g?G Ig a fixed graded ideal of R. We give a number of results concerning this class of graded submodules and their homogeneous components. A proper graded submodule N of M is said to be a graded Ie-prime submodule of M if whenever rg ? h(R) and mh ? h(M) with rgmh ? N ? IeN, then either rg ? (N :R M) or mh ? N.

scholarly journals φ-PRIME SUBMODULES

2009 ◽  
Vol 52 (2) ◽  
pp. 253-259 ◽  
Author(s):  
NASER ZAMANI
Keyword(s):  
Commutative Ring ◽  
Prime Submodules ◽  
Prime Submodule

AbstractLet R be a commutative ring with non-zero identity and M be a unitary R-module. Let (M) be the set of all submodules of M, and φ: (M) → (M) ∪ {∅} be a function. We say that a proper submodule P of M is a prime submodule relative to φ or φ-prime submodule if a ∈ R and x ∈ M, with ax ∈ P ∖ φ(P) implies that a ∈(P :RM) or x ∈ P. So if we take φ(N) = ∅ for each N ∈ (M), then a φ-prime submodule is exactly a prime submodule. Also if we consider φ(N) = {0} for each submodule N of M, then in this case a φ-prime submodule will be called a weak prime submodule. Some of the properties of this concept will be investigated. Some characterisations of φ-prime submodules will be given, and we show that under some assumptions prime submodules and φ1-prime submodules coincide.

S(0)-Weakly second submodules

2019 ◽  
Vol 12 (05) ◽  
pp. 1950072
Author(s):  
Indah Emilia Wijayanti ◽  
Dian Ariesta Yuwaningsih ◽  
Salmah
Keyword(s):  
Commutative Ring ◽  
Prime Submodules

We introduce the dual notions of [Formula: see text]-weakly prime submodules, that is, [Formula: see text]-weakly second submodules in a commutative ring with identity. We investigate the properties of [Formula: see text]-weakly second submodules and obtain some related useful characterizations of this dualization. Moreover, we also prove some properties of [Formula: see text]-weakly second submodules related to multiplication and comultiplication modules.

scholarly journals On 2-absorbing quasi primary submodules

Filomat ◽  
2017 ◽  
Vol 31 (10) ◽  
pp. 2943-2950 ◽  
Author(s):  
Suat Koc ◽  
Uregen Nagehan ◽  
Unsal Tekir
Keyword(s):  
Commutative Ring ◽  
Prime Submodules ◽  
Primary Submodules ◽  
Primary Submodule ◽  
Nonzero Identity

Let R be a commutative ring with nonzero identity, and let M be a nonzero unital R-module. In this article, we introduce the concept of 2-absorbing quasi primary submodules which is a generalization of prime submodules. We define 2-absorbing quasi primary submodule as a proper submodule N of M having the property that abm ? N, then ab ? ?(N :R M) or am ? radM(N) or bm ? radM(N): Various results and examples concerning 2-absorbing quasi primary submodules are given.

scholarly journals Approximaitly Prime Submodules and Some Related Concepts

2019 ◽  
Vol 32 (2) ◽  
pp. 103
Author(s):  
Ali Sh. Ajeel ◽  
Haibat K. Mohammad Ali
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Research Note ◽  
Prime Radical ◽  
Basic Properties ◽  
Prime Submodules ◽  
Prime Submodule ◽  
Definition Of

In this research note approximately prime submodules is defined as a new generalization of prime submodules of unitary modules over a commutative ring with identity. A proper submodule  of an -module  is called an approximaitly prime submodule of  (for short app-prime submodule), if when ever , where , , implies that either  or . So, an ideal  of a ring  is called app-prime ideal of  if   is an app-prime submodule of -module . Several basic properties, characterizations and examples of approximaitly prime submodules were given. Furthermore, the definition of approximaitly prime radical of submodules of modules were introduced, and some of it is properties were established.

scholarly journals $n$-absorbing and strongly $n$-absorbing second submodules

2021 ◽  
Vol 39 (1) ◽  
pp. 9-22
Author(s):  
Habibollah Ansari-Toroghy ◽  
Faranak Farshadifar ◽  
Sepideh ‎Maleki-Roudposhti
Keyword(s):  
Commutative Ring ◽  
Dual Notion

‎In this paper‎, ‎we introduce the concepts of $n$-absorbing and strongly $n$-absorbing second submodules as a dual notion of $n$-absorbing submodules of modules  over a commutative ring and obtain some related results‎. ‎In particular‎, ‎we investigate some results concerning strongly 2-absorbing second submodules‎.

scholarly journals Primary modules over commutative rings

2001 ◽  
Vol 43 (1) ◽  
pp. 103-111 ◽  
Author(s):  
Patrick F. Smith
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Prime Radical ◽  
Finitely Generated ◽  
One Dimensional ◽  
Prime Submodules ◽  
Primary Submodules ◽  
Commutative Domain

The radical of a module over a commutative ring is the intersection of all prime submodules. It is proved that if R is a commutative domain which is either Noetherian or a UFD then R is one-dimensional if and only if every (finitely generated) primary R-module has prime radical, and this holds precisely when every (finitely generated) R-module satisfies the radical formula for primary submodules.

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