scholarly journals Approximaitly Quasi-primary Submodules

2020 ◽  
Vol 33 (4) ◽  
pp. 92
Author(s):  
Ali Sh. Ajeel ◽  
Omar A. Abdulla ◽  
Haibat K. Mohammadali
Keyword(s):  
Commutative Ring ◽  
Left Module ◽  
Basic Properties ◽  
Primary Submodules

      In this paper, we introduce and study the notation of approximaitly quasi-primary submodules of a unitary left -module  over a commutative ring  with identity. This concept is a generalization of prime and primary submodules, where a proper submodule  of an -module  is called an approximaitly quasi-primary (for short App-qp) submodule of , if , for , , implies that either  or , for some . Many basic properties, examples and characterizations of this concept are introduced.

scholarly journals Approximaitly Primary Submodules

2019 ◽  
Vol 24 (5) ◽  
pp. 105
Author(s):  
Ali Sh. Ajeel ◽  
Haibat K. Mohammadali
Keyword(s):  
Positive Integer ◽  
Commutative Ring ◽  
The Other ◽  
Left Module ◽  
Basic Properties ◽  
Other Hand ◽  
Prime Submodules ◽  
Primary Submodules ◽  
Primary Submodule

The study deals with the notion of an approximaitly primary submodules of unitary left -module  over a commutative ring  with identity as a generalization of a primary submodules and approximaitly prime submodules, where a proper submodule  of an -module  is called an approximaitly primary submodule of , if whenever , for , , implies that either  or  for some positive integer  of . Several characterizations, basic properties of this concept are given. On the other hand the relationships of this concept with some classes of modules are studied. Furthermore, the behavior of approximaitly primary submodule under -homomorphism are discussed   http://dx.doi.org/10.25130/tjps.24.2019.098

scholarly journals P-Semi Hollow-Lifting Modules

2021 ◽  
Vol 17 (3) ◽  
pp. 37-48
Author(s):  
Mukdad Qaess Hussain ◽  
Darya Jabar Abdul Kareem
Keyword(s):  
Left Module ◽  
Basic Properties ◽  
Comprehensive Study

Let R be a ring with identity and Q be a unitary left Module over R. In this paper, we introduced the concept of p-semi hollow-lifting Module as generalization of semi hollow-lifting Module. Also, give a comprehensive study of basic properties of p-semi hollow-lifting Modules and some related concepts.

scholarly journals On Almost Semiprime Submodules

Algebra ◽  
2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Farkhonde Farzalipour
Keyword(s):  
Commutative Ring ◽  
Finitely Generated ◽  
Basic Properties ◽  
Nonzero Identity

We introduce the concept of almost semiprime submodules of unitary modules over a commutative ring with nonzero identity. We investigate some basic properties of almost semiprime and weakly semiprime submodules and give some characterizations of them, especially for (finitely generated faithful) multiplication modules.

Comaximal factorization of lifting ideals

2020 ◽  
pp. 2250044
Author(s):  
Esmaeil Rostami ◽  
Sina Hedayat ◽  
Reza Nekooei ◽  
Somayeh Karimzadeh
Keyword(s):  
Commutative Ring ◽  
Proper Ideal ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
New Approach ◽  
Ideal Formula ◽  
Basic Properties ◽  
Necessary And Sufficient

A proper ideal [Formula: see text] of a commutative ring [Formula: see text] is called lifting whenever idempotents of [Formula: see text] lift to idempotents of [Formula: see text]. In this paper, many of the basic properties of lifting ideals are studied and we prove and extend some well-known results concerning lifting ideals and lifting idempotents by a new approach. Furthermore, we give a necessary and sufficient condition for every proper ideal of a commutative ring to be a product of pairwise comaximal lifting ideals.

CAYLEY SUM GRAPHS OF IDEALS OF A COMMUTATIVE RING

2014 ◽  
Vol 96 (3) ◽  
pp. 289-302 ◽  
Author(s):  
M. AFKHAMI ◽  
Z. BARATI ◽  
K. KHASHYARMANESH ◽  
N. PAKNEJAD
Keyword(s):  
Commutative Ring ◽  
Directed Graph ◽  
Undirected Graph ◽  
Clique Number ◽  
Basic Properties ◽  
Vertex Set ◽  
Ring Graph ◽  
Genus One ◽  
Cayley Sum Graph

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}R$ be a commutative ring, $I(R)$ be the set of all ideals of $R$ and $S$ be a subset of $I^*(R)=I(R)\setminus \{0\}$. We define a Cayley sum digraph of ideals of $R$, denoted by $\overrightarrow{\mathrm{Cay}}^+ (I(R),S)$, as a directed graph whose vertex set is the set $I(R)$ and, for every two distinct vertices $I$ and $J$, there is an arc from $I$ to $J$, denoted by $I\longrightarrow J$, whenever $I+K=J$, for some ideal $K $ in $S$. Also, the Cayley sum graph $ \mathrm{Cay}^+ (I(R), S)$ is an undirected graph whose vertex set is the set $I(R)$ and two distinct vertices $I$ and $J$ are adjacent whenever $I+K=J$ or $J+K=I$, for some ideal $K $ in $ S$. In this paper, we study some basic properties of the graphs $\overrightarrow{\mathrm{Cay}}^+ (I(R),S)$ and $ \mathrm{Cay}^+ (I(R), S)$ such as connectivity, girth and clique number. Moreover, we investigate the planarity, outerplanarity and ring graph of $ \mathrm{Cay}^+ (I(R), S)$ and also we provide some characterization for rings $R$ whose Cayley sum graphs have genus one.

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 W-Closed Submodule and Related Concepts

2018 ◽  
Vol 31 (2) ◽  
pp. 164
Author(s):  
Haibat K. Mohammad Ali ◽  
Mohammad E. Dahsh
Keyword(s):  
Commutative Ring ◽  
Chain Condition ◽  
Essential Extension ◽  
Basic Properties ◽  
Closed Submodule ◽  
Closed Submodules

    Let R be a commutative ring with identity, and M be a left untial module. In this paper we introduce and study the concept w-closed submodules, that is stronger form of the concept of closed submodules, where asubmodule K of a module M is called w-closed in M, "if it has no proper weak essential extension in M", that is if there exists a submodule L of M with K is weak essential submodule of L then K=L. Some basic properties, examples of w-closed submodules are investigated, and some relationships between w-closed submodules and other related modules are studied. Furthermore, modules with chain condition on w-closed submodules are studied.   

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.

scholarly journals Graded near-rings

2016 ◽  
Vol 24 (1) ◽  
pp. 201-216
Author(s):  
Mariana Dumitru ◽  
Laura Năstăsescu ◽  
Bogdan Toader
Keyword(s):  
Commutative Ring ◽  
Constant Term ◽  
Graded Rings ◽  
Graded Ring ◽  
Basic Properties ◽  
Cancellative Monoid ◽  
Ring Of Polynomials ◽  
Left Cancellative Monoid

AbstractIn this paper, we consider graded near-rings over a monoid G as generalizations of graded rings over groups, and study some of their basic properties. We give some examples of graded near-rings having various interesting properties, and we define and study the Gop-graded ring associated to a G-graded abelian near-ring, where G is a left cancellative monoid and Gop is its opposite monoid. We also compute the graded ring associated to the graded near-ring of polynomials (over a commutative ring R) whose constant term is zero.

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