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.

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.

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 A note on GV-modules with Krull dimension

1990 ◽  
Vol 32 (3) ◽  
pp. 389-390 ◽  
Author(s):  
Dinh Huynh van ◽  
Patrick F. Smith ◽  
Robert Wisbauer
Keyword(s):  
Associative Ring ◽  
Simple Module ◽  
Krull Dimension ◽  
Finitely Generated ◽  
Direct Sums ◽  
Semisimple Modules ◽  
Factor Module ◽  
Maximal Submodule

AbstractExtending a result of Boyle and Goodearl in [1] on V-rings it was shown in Yousif [11] that a generalized V-module (GV-module) has Krull dimension if and only if it is noetherian. Our note is based on the observation that every GV-module has a maximal submodule (Lemma 1). Applying a theorem of Shock [6] we immediately obtain that a GV-module has acc on essential submodules if and only if for every essential submodule K ⊂ M the factor module M/K has finitely generated socle. Yousif's result is obtained as a corollary.Let R be an associative ring with unity and R-Mod the category of unital left R-modules. Soc M denotes the socle of an R-module M. If K ⊂ M is an essential submodule we write K⊴M.An R-module M is called co-semisimple or a V-module, if every simple module is M-injective ([2], [7], [9], [10]). According to Hirano [3] M is a generalized V-module or GV-module, if every singular simple R-module is M-injective. This extends the notion of a left GV-ring in Ramamurthi-Rangaswamy [5].It is easy to see that submodules, factor modules and direct sums of co-semisimple modules (GV-modules) are again co-semisimple (GV-modules) (e.g. [10, § 23]).

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.

On Minimax and Related Modules

1992 ◽  
Vol 44 (1) ◽  
pp. 154-166 ◽  
Author(s):  
Peter Rudlof
Keyword(s):  
Homomorphic Image ◽  
Finite Codimension ◽  
Noetherian Rings ◽  
Finitely Generated ◽  
Maximal Submodule ◽  
Minimax Modules ◽  
Minimax Module ◽  
Generalized Classes

AbstractA module M is called a minimax module, if it has a finitely generated submodule U such that M/U is Artinian. This paper investigates minimax modules and some generalized classes over commutative Noetherian rings. One of our main results is: M is minimax iff every decomposition of a homomorphic image of M is finite.From this we deduce that:- All couniform modules are minimax.- All modules of finite codimension are minimax.- Essential covers of minimax modules are minimax. With the aid of these corollaries we completely determine the structure of couniform modules and modules of finite codimension.We then examine the following variants of the minimax property:- replace U “ finitely generated” by U “ coatomic” (i.e. every proper submodule of U is contained in a maximal submodule);- replace M/U “ Artinian” by M/U “ semi-Artinian” (i.e. every proper submodule of M/U contains a minimal submodule).

Automorphisms of finite order of nilpotent groups II

2014 ◽  
Vol 51 (4) ◽  
pp. 547-555 ◽  
Author(s):  
B. Wehrfritz
Keyword(s):  
Nilpotent Group ◽  
Finite Order ◽  
Finite Index ◽  
Nilpotent Groups ◽  
Finitely Generated

Let G be a nilpotent group with finite abelian ranks (e.g. let G be a finitely generated nilpotent group) and suppose φ is an automorphism of G of finite order m. If γ and ψ denote the associated maps of G given by \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\gamma :g \mapsto g^{ - 1} \cdot g\phi and \psi :g \mapsto g \cdot g\phi \cdot g\phi ^2 \cdots \cdot \cdot g\phi ^{m - 1} for g \in G,$$ \end{document} then Gγ · kerγ and Gψ · ker ψ are both very large in that they contain subgroups of finite index in G.

Collectives of Automata in Finitely Generated Groups

2020 ◽  
Vol 108 (5-6) ◽  
pp. 671-678
Author(s):  
D. V. Gusev ◽  
I. A. Ivanov-Pogodaev ◽  
A. Ya. Kanel-Belov
Keyword(s):  
Finitely Generated ◽  
Finitely Generated Groups

scholarly journals TEST IDEALS IN RINGS WITH FINITELY GENERATED ANTI-CANONICAL ALGEBRAS – CORRIGENDUM

2016 ◽  
Vol 17 (4) ◽  
pp. 979-980
Author(s):  
Alberto Chiecchio ◽  
Florian Enescu ◽  
Lance Edward Miller ◽  
Karl Schwede
Keyword(s):  
Finitely Generated
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