Endomorphism Rings and Gabriel Topologies

1984 ◽  
Vol 36 (2) ◽  
pp. 193-205 ◽  
Author(s):  
Soumaya Makdissi Khuri
Keyword(s):  
Endomorphism Ring ◽  
Lattice Isomorphism ◽  
Close Connection ◽  
Projective Modules ◽  
Finitely Generated ◽  
Basic Tool ◽  
Endomorphism Rings ◽  
Finite Dimensional ◽  
Faithful Module ◽  
Correspondence Theorem

A basic tool in the usual presentation of the Morita theorems is the correspondence theorem for projective modules. Let RM be a left R-module and B = HomR(M, M). When M is a progenerator, there is a close connection (in fact a lattice isomorphism) between left R-submodules of M and left ideals of B, which can be applied to the solution of problems such as characterizing when the endomorphism ring of a finitely generated projective faithful module is simple or right Noetherian. More generally, Faith proved that this connection can be retained in suitably modified form when M is just a generator in R-mod ([4], [2], [3]). In this form the correspondence theorem can be applied to show, e.g., that, when RM is a generator, then (a): RM is finite-dimensional if and only if B is a left finite-dimensional ring and in this case d(RM) = d(BB), and (b): If RM is nonsingular then B is a left nonsingular ring ([6]).

scholarly journals QF-3 endomorphism rings of Σ-quasi-projective modules

1988 ◽  
Vol 30 (2) ◽  
pp. 215-220 ◽  
Author(s):  
José L. Gómez Pardo ◽  
Nieves Rodríguez González
Keyword(s):  
Endomorphism Ring ◽  
Projective Module ◽  
Sufficient Conditions ◽  
Torsion Theory ◽  
Projective Modules ◽  
Finitely Generated ◽  
Endomorphism Rings ◽  
Quotient Module ◽  
Qf Ring ◽  
Necessary And Sufficient

A ring R is called left QF-3 if it has a minimal faithful left R-module. The endomorphism ring of such a module has been recently studied in [7], where conditions are given for it to be a left PF ring or a QF ring. The object of the present paper is to study, more generally, when the endomorphism ring of a Σ-quasi-projective module over any ring R is left QF-3. Necessary and sufficient conditions for this to happen are given in Theorem 2. An useful concept in this investigation is that of a QF-3 module which has been introduced in [11]. If M is a finitely generated quasi-projective module and σ[M] denotes the category of all modules isomorphic to submodules of modules generated by M, then we show that End(RM) is a left QF-3 ring if and only if the quotient module of M modulo its torsion submodule (in the torsion theory of σ[M] canonically defined by M) is a QF-3 module (Corollary 4). Finally, we apply these results to the study of the endomorphism ring of a minimal faithful R-module over a left QF-3 ring, extending some of the results of [7].

scholarly journals Isomorphisms between endomorphism rings of projective modules

1993 ◽  
Vol 35 (3) ◽  
pp. 353-355 ◽  
Author(s):  
José Luis García ◽  
Juan Jacobo Simón
Keyword(s):  
Morita Equivalence ◽  
Projective Modules ◽  
Finitely Generated ◽  
Endomorphism Rings ◽  
Morita Equivalent ◽  
Free Modules

Let R and S be arbitrary rings, RM and SN countably generated free modules, and let φ:End(RM)→End(sN) be an isomorphism between the endomorphism rings of M and N. Camillo [3] showed in 1984 that these assumptions imply that R and S are Morita equivalent rings. Indeed, as Bolla pointed out in [2], in this case the isomorphism φ must be induced by some Morita equivalence between R and S. The same holds true if one assumes that RM and SN are, more generally, non-finitely generated free modules.

Modules Whose Endomorphism Rings Are (m, n)-Coherent

2019 ◽  
Vol 26 (02) ◽  
pp. 231-242
Author(s):  
Xiaoqiang Luo ◽  
Lixin Mao
Keyword(s):  
Left Ideal ◽  
Endomorphism Ring ◽  
Finitely Generated ◽  
Endomorphism Rings ◽  
Von Neumann ◽  
Von Neumann Regular ◽  
Coherent Ring ◽  
Left Annihilator

Let M be a right R-module with endomorphism ring S. We study the left (m, n)-coherence of S. It is shown that S is a left (m, n)-coherent ring if and only if the left annihilator [Formula: see text] is a finitely generated left ideal of Mn(S) for any M-m-generated submodule X of Mn if and only if every M-(n, m)-presented right R-module has an add M-preenvelope. As a consequence, we investigate when the endomorphism ring S is left coherent, left pseudo-coherent, left semihereditary or von Neumann regular.

scholarly journals A note on deformations of Gorenstein-projective modules over finite dimensional algebras

2019 ◽  
Vol 53 (supl) ◽  
pp. 245-256 ◽  
Author(s):  
José A. Vélez-Marulanda
Keyword(s):  
Projective Modules ◽  
Finitely Generated ◽  
Universal Deformation Rings ◽  
Finite Dimensional ◽  
Gorenstein Projective Modules ◽  
Deformation Rings ◽  
Finite Dimensional Algebras

In this note, we present a survey of results concerning universal deformation rings of finitely generated Gorenstein-projective modules over finite dimensional algebras.

On Hermitian endomorphism rings

2017 ◽  
Vol 16 (11) ◽  
pp. 1750220
Author(s):  
Wanru Zhang
Keyword(s):  
Endomorphism Ring ◽  
Necessary Conditions ◽  
Exchange Property ◽  
Projective Modules ◽  
Endomorphism Rings ◽  
Dual Problems ◽  
Injective Modules ◽  
Sufficient And Necessary Conditions

Let [Formula: see text] be a right [Formula: see text]-module with finite exchange property and let [Formula: see text] be its endomorphism ring. In this paper, some sufficient and necessary conditions for [Formula: see text] to be a Hermitian ring are given. Moreover, we investigate Hermitian endomorphism rings of quasi-projective modules by means of completions of diagrams. The dual problems for quasi-injective modules are also studied.

Endomorphism rings and the category of finitely generated projective modules

2000 ◽  
Vol 28 (8) ◽  
pp. 3837-3852 ◽  
Author(s):  
J.L. García ◽  
Martínez Hernández ◽  
P.L. Gómez Sánchez
Keyword(s):  
Projective Modules ◽  
Finitely Generated ◽  
Endomorphism Rings

CHAIN CONDITIONS ON NON-SUMMANDS

2011 ◽  
Vol 10 (03) ◽  
pp. 475-489 ◽  
Author(s):  
PINAR AYDOĞDU ◽  
A. ÇIĞDEM ÖZCAN ◽  
PATRICK F. SMITH
Keyword(s):  
Noetherian Ring ◽  
Jacobson Radical ◽  
Finitely Generated ◽  
Chain Conditions ◽  
Finite Dimensional

Let R be a ring. Modules satisfying ascending or descending chain conditions (respectively, acc and dcc) on non-summand submodules belongs to some particular classes [Formula: see text], such as the class of all R-modules, finitely generated, finite-dimensional and cyclic modules, are considered. It is proved that a module M satisfies acc (respectively, dcc) on non-summands if and only if M is semisimple or Noetherian (respectively, Artinian). Over a right Noetherian ring R, a right R-module M satisfies acc on finitely generated non-summands if and only if M satisfies acc on non-summands; a right R-module M satisfies dcc on finitely generated non-summands if and only if M is locally Artinian. Moreover, if a ring R satisfies dcc on cyclic non-summand right ideals, then R is a semiregular ring such that the Jacobson radical J is left t-nilpotent.

scholarly journals Generating infinite monoids of cellular automata

2021 ◽  
pp. 2250215
Author(s):  
Alonso Castillo-Ramirez
Keyword(s):  
Cellular Automata ◽  
Normal Subgroups ◽  
Finitely Generated ◽  
Residually Finite ◽  
Residually Finite Group ◽  
Group Of Units ◽  
Finite Dimensional ◽  
Index Condition ◽  
Indicable Group

For a group [Formula: see text] and a set [Formula: see text], let [Formula: see text] be the monoid of all cellular automata over [Formula: see text], and let [Formula: see text] be its group of units. By establishing a characterization of surjunctive groups in terms of the monoid [Formula: see text], we prove that the rank of [Formula: see text] (i.e. the smallest cardinality of a generating set) is equal to the rank of [Formula: see text] plus the relative rank of [Formula: see text] in [Formula: see text], and that the latter is infinite when [Formula: see text] has an infinite decreasing chain of normal subgroups of finite index, condition which is satisfied, for example, for any infinite residually finite group. Moreover, when [Formula: see text] is a vector space over a field [Formula: see text], we study the monoid [Formula: see text] of all linear cellular automata over [Formula: see text] and its group of units [Formula: see text]. We show that if [Formula: see text] is an indicable group and [Formula: see text] is finite-dimensional, then [Formula: see text] is not finitely generated; however, for any finitely generated indicable group [Formula: see text], the group [Formula: see text] is finitely generated if and only if [Formula: see text] is finite.

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