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]).

LOCALLY FINITELY GENERATED MODULES OVER RINGS WITH ENOUGH IDEMPOTENTS

2009 ◽  
Vol 08 (06) ◽  
pp. 885-901 ◽  
Author(s):  
LIDIA ANGELERI HÜGEL ◽  
JOSÉ ANTONIO DE LA PEÑA
Keyword(s):  
Finite Length ◽  
Associative Ring ◽  
Projective Modules ◽  
Finitely Generated ◽  
Direct Sums ◽  
Indecomposable Modules ◽  
Primitive Idempotents ◽  
Almost Split Sequences ◽  
Finitely Generated Modules ◽  
Infinite Set

Let R = ⊕ Reλ = ⊕ eλ R be an associative ring with enough idempotents indexed over a possibly infinite set Λ. Assume that {eλ : λ ∈ Λ} is a set of pairwise orthogonal primitive idempotents, and that R is locally bounded, that is, the projective modules eλR and Reλ are of finite length for each λ ∈ Λ. We prove the existence of almost split sequences ending at the indecomposable finitely generated non-projective unital R-modules. Moreover, we consider the unital R-modules X that are locally finitely generated, that is, Xeλ is a finitely generated eλ Reλ-module for all λ ∈ Λ. We show that such X accept perfect decompositionsX = ⊕ Xi as direct sums of indecomposable modules.

scholarly journals BOUNDED AND FULLY BOUNDED MODULES

2011 ◽  
Vol 84 (3) ◽  
pp. 433-440
Author(s):  
A. HAGHANY ◽  
M. MAZROOEI ◽  
M. R. VEDADI
Keyword(s):  
Krull Dimension ◽  
Finitely Generated ◽  
Morita Invariant ◽  
Prime Submodule ◽  
Invariant Properties ◽  
Noetherian Modules

AbstractGeneralizing the concept of right bounded rings, a module MR is called bounded if annR(M/N)≤eRR for all N≤eMR. The module MR is called fully bounded if (M/P) is bounded as a module over R/annR(M/P) for any ℒ2-prime submodule P◃MR. Boundedness and right boundedness are Morita invariant properties. Rings with all modules (fully) bounded are characterized, and it is proved that a ring R is right Artinian if and only if RR has Krull dimension, all R-modules are fully bounded and ideals of R are finitely generated as right ideals. For certain fully bounded ℒ2-Noetherian modules MR, it is shown that the Krull dimension of MR is at most equal to the classical Krull dimension of R when both dimensions exist.

scholarly journals A note on the fixed subring of an FPF ring

1989 ◽  
Vol 40 (1) ◽  
pp. 109-111 ◽  
Author(s):  
John Clark
Keyword(s):  
Finite Group ◽  
Associative Ring ◽  
Finitely Generated ◽  
Group Of Automorphisms ◽  
Direct Sum ◽  
Epimorphic Image ◽  
A Finite Group

An associative ring R with identity is called a left (right) FPF ring if given any finitely generated faithful left (right) R-module A and any left (right) R-module M then M is the epimorphic image of a direct sum of copies of A. Faith and Page have asked if the subring of elements fixed by a finite group of automorphisms of an FPF ring need also be FPF. Here we present examples showing the answer to be negative in general.

Q-Divisible Modules

1971 ◽  
Vol 14 (4) ◽  
pp. 491-494 ◽  
Author(s):  
Efraim P. Armendariz
Keyword(s):  
Quotient Ring ◽  
Torsion Class ◽  
Direct Sums ◽  
Factor Module ◽  
Left Quotient ◽  
Divisible Modules

Let R be a ring with 1 and let Q denote the maximal left quotient ring of R [6]. In a recent paper [12], Wei called a (left). R-module M divisible in case HomR (Q, N)≠0 for each nonzero factor module N of M. Modifying the terminology slightly we call such an R-module a Q-divisible R-module. As shown in [12], the class D of all Q-divisible modules is closed under factor modules, extensions, and direct sums and thus is a torsion class in the sense of Dickson [5].

Modules with annihilation property

2020 ◽  
pp. 2150126
Author(s):  
Rasul Mohammadi ◽  
Ahmad Moussavi ◽  
Masoome Zahiri
Keyword(s):  
Commutative Ring ◽  
Associative Ring ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Finitely Generated ◽  
Ideal Formula ◽  
Ordered Monoid ◽  
Zero Divisors ◽  
The Right ◽  
Totally Ordered Monoid

Let [Formula: see text] be an associative ring with identity. A right [Formula: see text]-module [Formula: see text] is said to have Property ([Formula: see text]), if each finitely generated ideal [Formula: see text] has a nonzero annihilator in [Formula: see text]. Evans [Zero divisors in Noetherian-like rings, Trans. Amer. Math. Soc. 155(2) (1971) 505–512.] proved that, over a commutative ring, zero-divisor modules have Property ([Formula: see text]). We study and construct various classes of modules with Property ([Formula: see text]). Following Anderson and Chun [McCoy modules and related modules over commutative rings, Comm. Algebra 45(6) (2017) 2593–2601.], we introduce [Formula: see text]-dual McCoy modules and show that, for every strictly totally ordered monoid [Formula: see text], faithful symmetric modules are [Formula: see text]-dual McCoy. We then use this notion to give a characterization for modules with Property ([Formula: see text]). For a faithful symmetric right [Formula: see text]-module [Formula: see text] and a strictly totally ordered monoid [Formula: see text], it is proved that the right [Formula: see text]-module [Formula: see text] is primal if and only if [Formula: see text] is primal with Property ([Formula: see text]).

Epimorphisms of Modules which must be Isomorphisms

1973 ◽  
Vol 16 (4) ◽  
pp. 513-515 ◽  
Author(s):  
D. Ž. Djoković
Keyword(s):  
Associative Ring ◽  
Finitely Generated

Let R be an associative ring (not necessarily with identity).R is a left П-ring if it has the following property: Let M be a finitely generated left R-module, N a submodule of M and ϕ:N→M an epimorphism. Then ϕ is an isomorphism.

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 Rings whose finitely generated modules are direct sums of cyclics

1974 ◽  
Vol 32 (1) ◽  
pp. 152-172 ◽  
Author(s):  
Thomas S Shores ◽  
Roger Wiegand
Keyword(s):  
Finitely Generated ◽  
Direct Sums ◽  
Finitely Generated Modules
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