scholarly journals Characterizations of unique maximal submodule and strong m-system

2021 ◽  
Keyword(s):  
Maximal Submodule

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.

Standardly stratified algebras and cellular algebras

2002 ◽  
Vol 133 (1) ◽  
pp. 37-53 ◽  
Author(s):  
CHANGCHANG XI
Keyword(s):  
Projective Cover ◽  
Complete List ◽  
Injective Envelope ◽  
Finite Chain ◽  
Artin Algebra ◽  
Simple Modules ◽  
Cellular Algebras ◽  
Maximal Submodule ◽  
Standard Module ◽  
Composition Factors

Let A be an Artin algebra. Then there are finitely many non-isomorphic simple A-modules. Suppose S1, S2, …, Sn form a complete list of all non-isomorphic simple A-modules and we fix this ordering of simple modules. Let Pi and Qi be the projective cover and the injective envelope of Si respectively. With this order of simple modules we define for each i the standard module Δ(i) to be the maximal quotient of Pi with composition factors Sj with j [les ] i. Let Δ be the set of all these standard modules Δ(i). We denote by [Fscr ](Δ) the subcategory of A-mod whose objects are the modules M which have a Δ-filtration, namely there is a finite chain0 = M0 ⊂ M1 ⊂ M2 ⊂ … ⊂ Mt = Mof submodules of M such that Mi/Mi−1 is isomorphic to a module in Δ for all i. The modules in [Fscr ](Δ) are called Δ-good modules. Dually, we define the costandard module ∇(i) to be the maximal submodule of Qi with composition factors Sj with j [les ] i and denote by ∇ the collection of all costandard modules. In this way, we have also the subcategory [Fscr ](∇) of A-mod whose objects are these modules which have a ∇-filtration. Of course, we have the notion of ∇-good modules. Note that Δ(n) is always projective and ∇(n) is always injective.

scholarly journals On quasi-duo rings

1995 ◽  
Vol 37 (1) ◽  
pp. 21-31 ◽  
Author(s):  
Hua-Ping Yu
Keyword(s):  
Natural Number ◽  
Left Ideal ◽  
Commutative Rings ◽  
Noncommutative Rings ◽  
Perfect Ring ◽  
Maximal Submodule ◽  
Bass Conjecture ◽  
Infinite Sets

Bass [1] proved that if R is a left perfect ring, then R contains no infinite sets of orthogonal idempotents and every nonzero left R-module has a maximal submodule, and asked if this property characterizes left perfect rings ([1], Remark (ii), p. 470). The fact that this is true for commutative rings was proved by Hamsher [12], and that this is not true in general was demonstrated by examples of Cozzens [7] and Koifman [14]. Hamsher's result for commutative rings has been extended to some noncommutative rings. Call a ring left duo if every left ideal is two-sided; Chandran [5] proved that Bass’ conjecture is true for left duo rings. Call a ring R weakly left duo if for every r ε R, there exists a natural number n(r) (depending on r) such that the principal left ideal Rrn(r) is two-sided. Recently, Xue [21] proved that Bass’ conjecture is still true for weakly left duo rings.

Rings over which each module possesses a maximal submodule

1997 ◽  
Vol 61 (3) ◽  
pp. 333-339 ◽  
Author(s):  
A. A. Tuganbaev
Keyword(s):  
Maximal Submodule
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