scholarly journals Semi-T-maximal sumodules

2019 ◽  
pp. 2725-2731
Author(s):  
Inaam M. A. Hadi ◽  
Alaa A. Elewi
Keyword(s):  
Commutative Ring ◽  
Semisimple Module ◽  
Maximal Submodule

Let  be a commutative ring with identity and  be an -module. In this work, we present the concept of semi--maximal sumodule as a generalization of -maximal submodule. We present that a submodule  of an -module  is a semi--maximal (sortly --max) submodule if  is a semisimple -module (where  is a submodule of ). We  investegate some properties of these kinds of modules.

The Intersection Property of Annihilators of Both Modules and Maximal Submodules Belonging to the Same Module

2020 ◽  
Vol 17 (2) ◽  
pp. 552-555
Author(s):  
Hatam Yahya Khalaf ◽  
Buthyna Nijad Shihab
Keyword(s):  
Commutative Ring ◽  
Intersection Property ◽  
Direct Sum ◽  
Unitary Module ◽  
Maximal Submodule ◽  
The Relationship

During that article T stands for a commutative ring with identity and that S stands for a unitary module over T. The intersection property of annihilatoers of a module X on a ring T and a maximal submodule W of M has been reviewed under this article where he provide several examples that explain that the property. Add to this a number of equivalent statements about the intersection property have been demonstrated as well as the direct sum of module that realize that the characteristic has studied here we proved that the modules that achieve the intersection property are closed under the direct sum with a specific condition. In addition to all this, the relationship between the modules that achieve the above characteristics with other types of modules has been given.

MODULES WITH COINDEPENDENT MAXIMAL SUBMODULES

2011 ◽  
Vol 10 (01) ◽  
pp. 73-99 ◽  
Author(s):  
PATRICK F. SMITH
Keyword(s):  
Positive Integer ◽  
Direct Summand ◽  
The Other ◽  
Property A ◽  
Other Hand ◽  
Semisimple Module ◽  
Maximal Submodule

Let R be a ring with identity. A unital left R-module M has the min-property provided the simple submodules of M are independent. On the other hand a left R-module M has the complete max-property provided the maximal submodules of M are completely coindependent, in other words every maximal submodule of M does not contain the intersection of the other maximal submodules of M. A semisimple module X has the min-property if and only if X does not contain distinct isomorphic simple submodules and this occurs if and only if X has the complete max-property. A left R-module M has the max-property if [Formula: see text] for every positive integer n and distinct maximal submodules L, Li (1 ≤ i ≤ n) of M. It is proved that a left R-module M has the complete max-property if and only if M has the max-property and every maximal submodule of M/Rad M is a direct summand, where Rad M denotes the radical of M, and in this case every maximal submodule of M is fully invariant. Various characterizations are given for when a module M has the max-property and when M has the complete max-property.

ON δ-LOCAL MODULES AND AMPLY δ-SUPPLEMENTED MODULES

2012 ◽  
Vol 12 (02) ◽  
pp. 1250144
Author(s):  
RACHID TRIBAK
Keyword(s):  
Finitely Generated ◽  
Semisimple Module ◽  
Maximal Submodule ◽  
Coatomic Module

The first part of this paper investigates the structure of δ-local modules. We prove that the following statements are equivalent for a module M: (i) M is δ-local; (ii) M is a coatomic module with either (a) M is a semisimple module having a maximal submodule N such that N is projective and M/N is singular, or (b) M has a unique essential maximal submodule K ≤ M such that for every maximal submodule L ≠ K, M/L is projective. The second part establishes some properties of finitely generated amply δ-supplemented modules.

scholarly journals Fuzzy Maximal Sub-Modules

2020 ◽  
pp. 1164-1172
Author(s):  
Maysoun A. Hamel ◽  
Hatam Y. Khalaf
Keyword(s):  
Basic Properties ◽  
Quotient Module ◽  
Semisimple Module ◽  
Maximal Submodule

In this paper, we introduce and study the notions of fuzzy quotient module, fuzzy (simple, semisimple) module and fuzzy maximal submodule. Also, we give many basic properties about these notions.

scholarly journals L-Hollow modules

2019 ◽  
Vol 24 (7) ◽  
pp. 104
Author(s):  
Thaer Z. Khlaif ◽  
Nada K. Abdullah
Keyword(s):  
Commutative Ring ◽  
Strong Form ◽  
Fundamental Properties ◽  
Maximal Submodule

To consider R is a commutative ring with unity,  be a nonzero unitary left   R-module,  is known hollow module if each proper submodule of  is small.  L-hollow module is a strong form of hollow module, where an R-module  is known L-hollow module if  has a unique maximal submodule which contains each small submodules. The current study deals with this class of modules and give several fundamental properties  related with this concept.   http://dx.doi.org/10.25130/tjps.24.2019.136

scholarly journals S-maximal Submodules

2015 ◽  
Vol 12 (1) ◽  
pp. 210-220
Author(s):  
Baghdad Science Journal
Keyword(s):  
Commutative Ring ◽  
Jacobson Radical ◽  
Maximal Submodule

Throughout this paper R represents a commutative ring with identity and all R-modules M are unitary left R-modules. In this work we introduce the notion of S-maximal submodules as a generalization of the class of maximal submodules, where a proper submodule N of an R-module M is called S-maximal, if whenever W is a semi essential submodule of M with N ? W ? M, implies that W = M. Various properties of an S-maximal submodule are considered, and we investigate some relationships between S-maximal submodules and some others related concepts such as almost maximal submodules and semimaximal submodules. Also, we study the behavior of S-maximal submodules in the class of multiplication modules. Farther more we give S-Jacobson radical of rings and modules. .

On the Maximal Spectrum of Semiprimitive Multiplication Modules

2008 ◽  
Vol 51 (3) ◽  
pp. 439-447
Author(s):  
Karim Samei
Keyword(s):  
Finite Number ◽  
Commutative Ring ◽  
Discrete Space ◽  
Multiplication Module ◽  
Prime Submodules ◽  
Maximal Submodule ◽  
Isolated Points ◽  
Maximal Spectrum ◽  
One Point Compactification

AbstractAnR-moduleMis called a multiplication module if for each submoduleNofM,N=IMfor some idealIofR. As defined for a commutative ringR, anR-moduleMis said to be semiprimitive if the intersection of maximal submodules ofMis zero. The maximal spectra of a semiprimitive multiplication moduleMare studied. The isolated points of Max(M) are characterized algebraically. The relationships among the maximal spectra ofM, Soc(M) and Ass(M) are studied. It is shown that Soc(M) is exactly the set of all elements ofMwhich belongs to every maximal submodule ofMexcept for a finite number. If Max(M) is infinite, Max(M) is a one-point compactification of a discrete space if and only ifMis Gelfand and for some maximal submoduleK, Soc(M) is the intersection of all prime submodules ofMcontained inK. WhenMis a semiprimitive Gelfand module, we prove that every intersection of essential submodules ofMis an essential submodule if and only if Max(M) is an almost discrete space. The set of uniform submodules ofMand the set of minimal submodules ofMcoincide. Ann(Soc(M))Mis a summand submodule ofMif and only if Max(M) is the union of two disjoint open subspacesAandN, whereAis almost discrete andNis dense in itself. In particular, Ann(Soc(M)) = Ann(M) if and only if Max(M) is almost discrete.

On quasi-radical of near-ring of polynomials

2019 ◽  
Vol 56 (2) ◽  
pp. 252-259
Author(s):  
Ebrahim Hashemi ◽  
Fatemeh Shokuhifar ◽  
Abdollah Alhevaz
Keyword(s):  
Commutative Ring ◽  
Nilpotent Elements ◽  
Ring Of Polynomials

Abstract The intersection of all maximal right ideals of a near-ring N is called the quasi-radical of N. In this paper, first we show that the quasi-radical of the zero-symmetric near-ring of polynomials R0[x] equals to the set of all nilpotent elements of R0[x], when R is a commutative ring with Nil (R)2 = 0. Then we show that the quasi-radical of R0[x] is a subset of the intersection of all maximal left ideals of R0[x]. Also, we give an example to show that for some commutative ring R the quasi-radical of R0[x] coincides with the intersection of all maximal left ideals of R0[x]. Moreover, we prove that the quasi-radical of R0[x] is the greatest quasi-regular (right) ideal of it.

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.

Derivations of differentially semiprime rings

2019 ◽  
Vol 12 (05) ◽  
pp. 1950079
Author(s):  
Ahmad Al Khalaf ◽  
Iman Taha ◽  
Orest D. Artemovych ◽  
Abdullah Aljouiiee
Keyword(s):  
Commutative Ring ◽  
Semiprime Ring ◽  
Prime Rings ◽  
Commutative Case ◽  
Lie Ring ◽  
Lie Rings ◽  
Semiprime Rings ◽  
Torsion Free

Earlier D. A. Jordan, C. R. Jordan and D. S. Passman have investigated the properties of Lie rings Der [Formula: see text] of derivations in a commutative differentially prime rings [Formula: see text]. We study Lie rings Der [Formula: see text] in the non-commutative case and prove that if [Formula: see text] is a [Formula: see text]-torsion-free [Formula: see text]-semiprime ring, then [Formula: see text] is a semiprime Lie ring or [Formula: see text] is a commutative ring.

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