Pedro A Guil Asensio - Academia.edu (original) (raw)
Papers by Pedro A Guil Asensio
Contemporary mathematics, 2006
arXiv (Cornell University), Mar 14, 2017
In this paper we study the Schröder-Bernstein problem for modules. We obtain a positive solution ... more In this paper we study the Schröder-Bernstein problem for modules. We obtain a positive solution for the Schröder-Bernstein problem for modules invariant under endomorphisms of their general envelopes under some mild conditions that are always satisfied, for example, in the case of injective, pureinjective or cotorsion envelopes. In the particular cases of injective envelopes and pure-injective envelopes, we are able to extend it further and we show that the Schröder-Bernstein problem has a positive solution even for modules that are invariant only under automorphisms of their injective envelopes or pure-injective envelopes. 2010 Mathematics Subject Classification. 16D40, 16D80. Key words and phrases. automorphism-invariant modules, endomorphism-invariant modules, envelopes.
Journal of Algebra and Its Applications, May 16, 2013
We show that a left CF ring is left artinian if and only if it is von Neumann regular modulo its ... more We show that a left CF ring is left artinian if and only if it is von Neumann regular modulo its left singular ideal. We deduce that a left FGF is Quasi-Frobenius (QF) under this assumption. This clarifies the role played by the Jacobson radical and the singular left ideal in the FGF and CF conjectures. In Sec. 3 of the paper, we study the structure of left artinian left CF rings. We prove that they are left continuous and left CEP rings.
Cambridge University Press eBooks, Mar 18, 2021
Contemporary mathematics, 2015
A module is called automorphism-invariant if it is invariant under any automorphism of its inject... more A module is called automorphism-invariant if it is invariant under any automorphism of its injective envelope. In this survey article we present the current state of art dealing with such class of modules. 1. Introduction. Johnson and Wong [20] proved that a module M is invariant under any endomorphism of its injective envelope if and only if any homomorphism from a submodule of M to M can be extended to an endomorphism of M. A module satisfying any of the above mentioned equivalent conditions is called a quasi-injective module. Clearly any injective module is quasi-injective. Most of the attempts of generalizing notions of injectivity or quasi-injectivity have focussed on relaxing conditions of lifting property of homomorphisms. For example, a module M was called pseudoinjective by Jain et al in [19] if every monomorphism from a submodule of M to M extends to an endomorphism of M (see [4], [9]). Dickson and Fuller were first to generalize the other aspect of quasi-injective modules that these are precisely the modules that are invariant under endomorphisms of their injective envelope. Dickson and Fuller studied modules that are invariant under automorphisms of their injective envelopes in [10] for the particular case of finite-dimensional algebras over fields F with more than two elements. But recently this notion has been studied for modules over any ring. A module M which is invariant under automorphisms of its injective envelope has been called an automorphism-invariant module in [24]. The dual notion has been defined in [30]. Let M be an automorphism-invariant module and M = A ⊕ B.
Tsukuba journal of mathematics, Dec 1, 1995
The problem of flnding the reflexive modules over generalizations of QFQFQF rings (and, in particul... more The problem of flnding the reflexive modules over generalizations of QFQFQF rings (and, in particular, over QF-3 rings) has a long tradition. One of the flrst contributions is due to Morita [10], who determined the flnitely generated reflexive modules over a right artinian QF-3 ring and, some years later, Masaike [8] extended this result by giving a characterization of reflexive modules over QF-3 rings with ACC (or DCC) on left annihilators. On the other hand, M\"uller [11] proved that if RUsRU_{s}RUs is a bimodule that induces a Morita duality, then the U-reflexive modules are precisely the linearly compact modules and this applies,
Journal of Algebra, Dec 1, 1997
Journal of Algebra, Oct 1, 2000
arXiv (Cornell University), Feb 2, 2013
A module is called automorphism-invariant if it is invariant under any automorphism of its inject... more A module is called automorphism-invariant if it is invariant under any automorphism of its injective hull. Dickson and Fuller have shown that if R is a finite-dimensional algebra over a field F with more than two elements then an indecomposable automorphism-invariant right R-module must be quasi-injective. In this note, we extend and simplify the proof of this result by showing that any automorphism-invariant module over an algebra over a field with more than two elements is quasi-injective. Our proof is based on the study of the additive unit structure of endomorphism rings.
arXiv (Cornell University), Oct 29, 2020
We prove that if u : K → M is a left minimal extension, then there exists an isomorphism between ... more We prove that if u : K → M is a left minimal extension, then there exists an isomorphism between two subrings, End M R (K) and End K R (M) of End R (K) and End R (M) respectively, modulo their Jacobson radicals. This isomorphism is used to deduce properties of the endomorphism ring of K from those of the endomorphism ring of M in certain situations such us when K is invariant under endomorphisms of M, or when K is invariant under automorphisms of M .
Contemporary mathematics, 2009
Publicacions Matematiques, Jul 1, 2023
arXiv (Cornell University), Jul 21, 2016
We give sufficient conditions to ensure that the ideal Φ(E) of E-phantom maps in a locally λ-pres... more We give sufficient conditions to ensure that the ideal Φ(E) of E-phantom maps in a locally λ-presentable exact category (A, E) is (special) (pre)covering ideal, where E is an exact substructure of (A, E). As a byproduct, we infer the existence of various covering ideals in categories of sheaves which have a meaningful geometrical motivation. In particular we deal with a Zariski-local notion of phantom maps in categories of sheaves. We would like to point out that our approach is necessarily different from [FGHT13], as the categories involved in most of the examples we are interested in do not have enough projective morphisms.
Journal of Algebra, Feb 1, 2014
Let R be a left hereditary ring. We show that if the left cotorsion envelope C (R R) of R is coun... more Let R be a left hereditary ring. We show that if the left cotorsion envelope C (R R) of R is countably generated, then R is a semilocal ring. In particular, we deduce that C (R R) is finitely generated if and only if R is a semiperfect cotorsion ring. Our proof is based on set theoretical counting arguments. We also discuss some possible extensions of this result.
Proceedings of the Edinburgh Mathematical Society, Jun 1, 1998
Let M be an essentially finitely generated injective (or, more generally, quasi-continuous) modul... more Let M be an essentially finitely generated injective (or, more generally, quasi-continuous) module. It is shown that if Af satisfies a mild uniqueness condition on essential closures of certain submodules, then the existence of an infinite independent set of submodules of M implies the existence of a larger independent set on some quotient of M modulo a directed union of direct summands. This provides new characterisations of injective (or quasi-continuous) modules of finite Goldie dimension. These results are then applied to the study of indecomposable decompositions of quasi-continuous modules and nonsingular CS modules.
Communications in Algebra, 1990
Journal of Algebra, Sep 1, 2010
It is well-known that a countably injective module is Σ-injective.
Proceedings of the American Mathematical Society, Aug 1, 1996
Let R be a ring, E = E(R R) its injective envelope, S = End(E R) and J the Jacobson radical of S.... more Let R be a ring, E = E(R R) its injective envelope, S = End(E R) and J the Jacobson radical of S. It is shown that if every finitely generated submodule of E embeds in a finitely presented module of projective dimension ≤ 1, then every finitley generated right S/J-module X is canonically isomorphic to Hom R (E, X ⊗ S E). This fact, together with a well-known theorem of Osofsky, allows us to prove that if, moreover, E/JE is completely pure-injective (a property that holds, for example, when the right pure global dimension of R is ≤ 1 and hence when R is a countable ring), then S is semiperfect and R R is finite-dimensional. We obtain several applications and a characterization of right hereditary right noetherian rings.
We give sufficient conditions to ensure that the ideal Φ(E) of E-phantom maps in a locally λ-pres... more We give sufficient conditions to ensure that the ideal Φ(E) of E-phantom maps in a locally λ-presentable exact category (A, E) is (special) (pre)covering ideal, where E is an exact substructure of (A, E). As a byproduct, we infer the existence of various covering ideals in categories of sheaves which have a meaningful geometrical motivation. In particular we deal with a Zariski-local notion of phantom maps in categories of sheaves. We would like to point out that our approach is necessarily different from [FGHT13], as the categories involved in most of the examples we are interested in do not have enough projective morphisms.
Contemporary mathematics, 2006
arXiv (Cornell University), Mar 14, 2017
In this paper we study the Schröder-Bernstein problem for modules. We obtain a positive solution ... more In this paper we study the Schröder-Bernstein problem for modules. We obtain a positive solution for the Schröder-Bernstein problem for modules invariant under endomorphisms of their general envelopes under some mild conditions that are always satisfied, for example, in the case of injective, pureinjective or cotorsion envelopes. In the particular cases of injective envelopes and pure-injective envelopes, we are able to extend it further and we show that the Schröder-Bernstein problem has a positive solution even for modules that are invariant only under automorphisms of their injective envelopes or pure-injective envelopes. 2010 Mathematics Subject Classification. 16D40, 16D80. Key words and phrases. automorphism-invariant modules, endomorphism-invariant modules, envelopes.
Journal of Algebra and Its Applications, May 16, 2013
We show that a left CF ring is left artinian if and only if it is von Neumann regular modulo its ... more We show that a left CF ring is left artinian if and only if it is von Neumann regular modulo its left singular ideal. We deduce that a left FGF is Quasi-Frobenius (QF) under this assumption. This clarifies the role played by the Jacobson radical and the singular left ideal in the FGF and CF conjectures. In Sec. 3 of the paper, we study the structure of left artinian left CF rings. We prove that they are left continuous and left CEP rings.
Cambridge University Press eBooks, Mar 18, 2021
Contemporary mathematics, 2015
A module is called automorphism-invariant if it is invariant under any automorphism of its inject... more A module is called automorphism-invariant if it is invariant under any automorphism of its injective envelope. In this survey article we present the current state of art dealing with such class of modules. 1. Introduction. Johnson and Wong [20] proved that a module M is invariant under any endomorphism of its injective envelope if and only if any homomorphism from a submodule of M to M can be extended to an endomorphism of M. A module satisfying any of the above mentioned equivalent conditions is called a quasi-injective module. Clearly any injective module is quasi-injective. Most of the attempts of generalizing notions of injectivity or quasi-injectivity have focussed on relaxing conditions of lifting property of homomorphisms. For example, a module M was called pseudoinjective by Jain et al in [19] if every monomorphism from a submodule of M to M extends to an endomorphism of M (see [4], [9]). Dickson and Fuller were first to generalize the other aspect of quasi-injective modules that these are precisely the modules that are invariant under endomorphisms of their injective envelope. Dickson and Fuller studied modules that are invariant under automorphisms of their injective envelopes in [10] for the particular case of finite-dimensional algebras over fields F with more than two elements. But recently this notion has been studied for modules over any ring. A module M which is invariant under automorphisms of its injective envelope has been called an automorphism-invariant module in [24]. The dual notion has been defined in [30]. Let M be an automorphism-invariant module and M = A ⊕ B.
Tsukuba journal of mathematics, Dec 1, 1995
The problem of flnding the reflexive modules over generalizations of QFQFQF rings (and, in particul... more The problem of flnding the reflexive modules over generalizations of QFQFQF rings (and, in particular, over QF-3 rings) has a long tradition. One of the flrst contributions is due to Morita [10], who determined the flnitely generated reflexive modules over a right artinian QF-3 ring and, some years later, Masaike [8] extended this result by giving a characterization of reflexive modules over QF-3 rings with ACC (or DCC) on left annihilators. On the other hand, M\"uller [11] proved that if RUsRU_{s}RUs is a bimodule that induces a Morita duality, then the U-reflexive modules are precisely the linearly compact modules and this applies,
Journal of Algebra, Dec 1, 1997
Journal of Algebra, Oct 1, 2000
arXiv (Cornell University), Feb 2, 2013
A module is called automorphism-invariant if it is invariant under any automorphism of its inject... more A module is called automorphism-invariant if it is invariant under any automorphism of its injective hull. Dickson and Fuller have shown that if R is a finite-dimensional algebra over a field F with more than two elements then an indecomposable automorphism-invariant right R-module must be quasi-injective. In this note, we extend and simplify the proof of this result by showing that any automorphism-invariant module over an algebra over a field with more than two elements is quasi-injective. Our proof is based on the study of the additive unit structure of endomorphism rings.
arXiv (Cornell University), Oct 29, 2020
We prove that if u : K → M is a left minimal extension, then there exists an isomorphism between ... more We prove that if u : K → M is a left minimal extension, then there exists an isomorphism between two subrings, End M R (K) and End K R (M) of End R (K) and End R (M) respectively, modulo their Jacobson radicals. This isomorphism is used to deduce properties of the endomorphism ring of K from those of the endomorphism ring of M in certain situations such us when K is invariant under endomorphisms of M, or when K is invariant under automorphisms of M .
Contemporary mathematics, 2009
Publicacions Matematiques, Jul 1, 2023
arXiv (Cornell University), Jul 21, 2016
We give sufficient conditions to ensure that the ideal Φ(E) of E-phantom maps in a locally λ-pres... more We give sufficient conditions to ensure that the ideal Φ(E) of E-phantom maps in a locally λ-presentable exact category (A, E) is (special) (pre)covering ideal, where E is an exact substructure of (A, E). As a byproduct, we infer the existence of various covering ideals in categories of sheaves which have a meaningful geometrical motivation. In particular we deal with a Zariski-local notion of phantom maps in categories of sheaves. We would like to point out that our approach is necessarily different from [FGHT13], as the categories involved in most of the examples we are interested in do not have enough projective morphisms.
Journal of Algebra, Feb 1, 2014
Let R be a left hereditary ring. We show that if the left cotorsion envelope C (R R) of R is coun... more Let R be a left hereditary ring. We show that if the left cotorsion envelope C (R R) of R is countably generated, then R is a semilocal ring. In particular, we deduce that C (R R) is finitely generated if and only if R is a semiperfect cotorsion ring. Our proof is based on set theoretical counting arguments. We also discuss some possible extensions of this result.
Proceedings of the Edinburgh Mathematical Society, Jun 1, 1998
Let M be an essentially finitely generated injective (or, more generally, quasi-continuous) modul... more Let M be an essentially finitely generated injective (or, more generally, quasi-continuous) module. It is shown that if Af satisfies a mild uniqueness condition on essential closures of certain submodules, then the existence of an infinite independent set of submodules of M implies the existence of a larger independent set on some quotient of M modulo a directed union of direct summands. This provides new characterisations of injective (or quasi-continuous) modules of finite Goldie dimension. These results are then applied to the study of indecomposable decompositions of quasi-continuous modules and nonsingular CS modules.
Communications in Algebra, 1990
Journal of Algebra, Sep 1, 2010
It is well-known that a countably injective module is Σ-injective.
Proceedings of the American Mathematical Society, Aug 1, 1996
Let R be a ring, E = E(R R) its injective envelope, S = End(E R) and J the Jacobson radical of S.... more Let R be a ring, E = E(R R) its injective envelope, S = End(E R) and J the Jacobson radical of S. It is shown that if every finitely generated submodule of E embeds in a finitely presented module of projective dimension ≤ 1, then every finitley generated right S/J-module X is canonically isomorphic to Hom R (E, X ⊗ S E). This fact, together with a well-known theorem of Osofsky, allows us to prove that if, moreover, E/JE is completely pure-injective (a property that holds, for example, when the right pure global dimension of R is ≤ 1 and hence when R is a countable ring), then S is semiperfect and R R is finite-dimensional. We obtain several applications and a characterization of right hereditary right noetherian rings.
We give sufficient conditions to ensure that the ideal Φ(E) of E-phantom maps in a locally λ-pres... more We give sufficient conditions to ensure that the ideal Φ(E) of E-phantom maps in a locally λ-presentable exact category (A, E) is (special) (pre)covering ideal, where E is an exact substructure of (A, E). As a byproduct, we infer the existence of various covering ideals in categories of sheaves which have a meaningful geometrical motivation. In particular we deal with a Zariski-local notion of phantom maps in categories of sheaves. We would like to point out that our approach is necessarily different from [FGHT13], as the categories involved in most of the examples we are interested in do not have enough projective morphisms.