del Río - Research

Ángel del Río Mateos

Departamento de Matemáticas

Facultad de Matemáticas

Universidad de Murcia

Murcia 30100, España

adelrio@um.es

(34) 868 88 35 37

Research

My papers Most of my research deals with problems connecting Ring Theory and Group Theory.
The papers are listed in decreasing order of appearance.

Á. del Río, Manuel Ruiz y Pavel Zalesskii
Subgroup separability in integral group rings
Journal of Algebra. DOI: 10.1016/j.jalgebra.2011.09.012
We provide a familly of finite groups which contains all the finite groups for which the group of units of ZG, the group ring of G with integral coefficientes, is subgroup separable. For most of this groups the mentioned property holds, however we have not been able to settle the question for some of the groups.

Eric Jespers, Gabriela Olteanu and Á. del Río
Rational group algebras of finite groups: from idempotents to units of integral group rings
Algebras and Representation Theory. DOI: 10.1007/s10468-010-9244-4
We give an explicit and character-free construction of a complete set of orthogonal primitive idempotents of a rational group algebra of a finite nilpotent group and a full description of the Wedderburn decomposition of such algebras. An immediate consequence is a well-known result of Roquette on the Schur indices of the simple components of group algebras of finite nilpotent groups. As an application, we obtain that the unit group of the integral group ring $\Z G$ of a finite nilpotent group $G$ has a subgroup of finite index that is generated by three nilpotent groups for which we have an explicit description of their generators. Another application is a new construction of free subgroups in the unit group. In all the constructions dealt with, pairs of subgroups (H,K), called strong Shoda pairs, and explicit constructed central elements e(G,H,K) play a crucial role. For arbitrary finite groups we prove that the primitive central idempotents of the rational group algebras are rational linear combinations of such e(G,H,K), with (H,K) strong Shoda pairs in subgroups of G.

Jairo Z. Gonçalvez and Á. del Río
Bass cyclic units as factors in a free group in integral group ring units
International Journal of Algebra and Computations, 21 (2011) 531–545. DOI: 10.1142/S0218196711006327
We prove that if u is a Bass cyclic unit of an integral group ring ZG of a solvable and finite group G, such that u has infinite order modulo the centre of U(ZG) and it is based on an element of prime order, then there is a non-abelian free group generated by a power of u and a power of a unit in G which is either a Bass cyclic unit or a bicyclic unit.

José J. Bernal, Á. del Río and J.J. Simón
Group code structures of affine-invariant codes
Journal of Algebra 325 (2011) 269-281. DOI:10.1016/j.jalgebra.2010.08.021
We describe all the group code structures of an affine-invariant code of length p^m in terms of a family of maps from F_{p^m} to the group of automorphisms of (F_{p^m},+). We also present a familly of non-obvious group code structures in an arbitrary affine-invariant code.

Ferran Cedó, E. Jespers and Á. del Río
Involutive Yang-Baxter Groups
Trans. of the Amer. Math. Soc. 362 (2010), no. 5, 2541–2558. DOI: 10.1090/S0002-9947-09-04927-7
In 1992 Drinfeld propose to classify the solutions of the set-theretical Yang-Baxter equation. Gateva-Ivanova y Van der Bergh, and Etingof, Schedler and Soloviev, have shown that the non-degenerate involutive solutions are in one-to-one correspondence with subgroups of the semidirect product of a free abelian group and the symmetric group on the rank of the group, acting in the obvious way, for which the projection onto the first component is a bijection. We study the groups obtained by projecting these group onto the second component. They are solvable. We obtain some results supprting the conjectura the every solvable group is of that type. Classifying such groups would provide a strategy to complete Drinfeld proposal.

Allen Herman, Gabriela Olteanu y Á. del Río
The gap between the Schur group and the subgroup generated by cyclic cyclotomic algebras
Israel J. Math. 176 (2010), 401–417. DOI: 10.1007/s11856-010-0034-9
Let K be an abelian extension of the rationals. Let S(K) be the Schur group of K and let CC(K) be the subgroup of S(K) generated by classes containing cyclic cyclotomic algebras. We characterize when CC(K) has finite index in S(K) in terms of the relative position of K in the lattice of cyclotomic extensions of the rationals.

José J. Bernal, Á. del Río and J.J. Simón
There are not non-obvious cyclic affine-invariant codes
``Applied algebra, Algebraic algorithms, and Error Correcting Codes, 2009 Proceedings'', Lecture Notes in Computer Science 5527 (2009) 101-106. DOI: 10.1007/978-3-642-02181-7_11
We show that an affine-invariant code C of length p^m is not permutation equivalent to a cyclic code except in the obvious cases: m=1 or C is either {0}, the repetition code or its dual.

José J. Bernal, Á. del Río and J.J. Simón
An intrinsical description of group codes
Designs, Codes and Cryptography 51, nº 3 (2009) 289-300. DOI: 10.1016/j.jsc.2007.07.019
A (left) group code of length n is a linear code which is the image of a (left) ideal of a group algebra via an isomorphism from F G to F^n which maps G to the standard basis of F^n. Many classical linear codes have been shown to be group codes. In this paper we obtain a criterion to decide when a linear code is a group code in terms of its intrinsical properties in the ambient space F^n, which does not assume an ``a priori'' group algebra structure on F^n. As an application we provide a family of groups (including metacyclic groups) for which every two-sided group code is an abelian group code. It is well known that Reed-Solomon codes are cyclic and its parity check extensions are elementary abelian group codes. These two classes of codes are included in the class of Cauchy codes. Using our criterion we classify the Cauchy codes of some lengths which are left group codes and the possible group code structures on these codes.

Allen Herman, Gabriela Olteanu y Á. del Río
The Schur group of an abelian number field
Journal of Pure and Applied Algebra 213 (2009), 22-33. DOI: 10.1016/j.jpaa.2008.05.002
We characterize the maximum r-local index of a Schur algebra over an abelian number field K in terms of global information determined by the field K, for r an arbitrary rational prime. This completes and unifies previous results of Janusz and Pendergrass.

Allen Herman, Gabriela Olteanu y Á. del Río
Ring isomorphism of cyclic cyclotomic algebras
Algebras and Representation Theory, 12 (2009) 265-370. DOI: 10.1007/s10468-009-9158-1
It is shown that ring isomorphism between cyclic cyclotomic algebras over cyclotomic number fields is essentially determined by the list of local Schur indices at all rational primes. As a consequence, ring isomorphism between simple components of the rational group algebras of finite metacyclic groups is determined by the center, the dimension over Q, and the list of local Schur indices at rational primes. An example is given to show that this does not hold for finite groups in general.

Gabriela Olteanu and Á. del Río
An algorithm to compute the Wedderburn decomposition of semisimple group algebras implemented in the GAP package wedderga
Journal of Symbolic Computation, 44 (2009) 507--516. DOI:10.1016/j.jsc.2007.07.019
We present an algorithm to compute the Wedderburn decomposition of arbitrary semisimple group algebras based on a computational approach of the Brauer-Witt theorem provided in [G. Olteanu, Computing theWedderburn decomposition of group algebras by the Brauer–Witt theorem, Math. Comp. 76 (2007) 1073–1087]. This method uses and extends the methods from the two previous papers. The algorithm was implemented in the GAP package wedderga.

Jairo Z. Gonçalves y Á. del Río
Bicyclic unit, Bass cyclic units and free groups
Journal of Group Theory 11 (2008) 247-265. DOI: 10.1515/JGT.2008.014
We provide necessary and sufficient conditions for a pair of bicyclic units generate a free group (non-abelian). We also prove that if G is a non-abelian group of order coprime with 6 then ZG contains a bicyclic unit b and a Bass cyclic unit u such that u and a power of b generate a free group.

Michael Dokuchaev, Á. del Río y Juan Jacobo Simón
Globalizations of partial actions on non unital rings
Proc. Amer. Math. Soc. 135 (2007) 343-352. DOI: 10.1090/S0002-9939-06-08503-0
We obtain a characterization of when a partial action of a group on a s-unital ring has a globalization.

Osnel Broche Cristo, Alexander Konovalov, Aurora Olivieri, Gabriela Olteanu y Á. del Río
wedderga
Accepted package of GAP - Groups, Algorithms, Programming - a System for Computational Discrete Algebra
The name of the package Wedderga stands for Wedderburn decomposition of group algebras. This is a GAP package to compute the simple components of the Wedderburn decomposition of semisimple group algebras of finite groups over abelian number fields and over finite fields. It also contains functions that compute the primitive central idempotents of the same kind of group algebras, and to construct crossed products over a group with coefficients in an associative ring with identity and the multiplication determined by a given action and twisting.

Eli Aljadeff y Á. del Río
Every projective Schur algebra is Brauer equivalent to a radical abelian algebra
Bulletin of the London Mathematica Society, 39 (2007) 731-740. DOI: 10.1112/blms/bdm056
We show that each central simple K-algebra generated by a group which is finite module K, is Brauer equivalent to radical abelian algebra, that is a crossed product (L/K,t), where L/K is a radical extension and the values taken by the cocycle t are finite module the K.

Gabriela Olteanu y Á. del Río
Group algebras of Kleinian type and groups of units
Journal of Algebra 318 n 2 (2007) 856-870. DOI: 10.1016/j.jalgebra.2007.03.026
We classify the Schur algebras of Kleinian type and the group algebras of Kleinian type. As an application, we characterize the group rings RG, with R an order in a number field and G a finite group, such that the group of units of RG is virtually a direct product of free-by-free groups.

Eric Jespers, Antonio Pita, Á. del Río, Manuel Ruiz y Pavel Zalesskii
Groups of units of integral group rings commensurable with direct products of free-by-free groups
Advances in Mathematics 212 nº 2 (2007) 692-722. DOI: 10.1016/j.aim.2006.11.005
We classify the finite groups of Kleinian type as the epimorphic images of the groups of a given list and we prove that a finite group G is of Kleinian type if and only if the group of units of ZG is commensuable with a direct product of free-by-free groups.

Osnel Broche Cristo y Á. del Río
Wedderburn decomposition of finite group algebras
Finite Fields and Their Applications 13 (2007) 71-79. DOI: 10.1016/j.ffa.2005.08.002
One shows how to generalized to semisimple finite group algebras the results from On monomial characters and central idempotents of rational group algebras.

Á. del Río y Juan Jacobo Simón
Finiteness conditions and infinite matrix rings
Proc. Amer. Math. Soc. 134 (2006) 1257-1263. DOI: 10.1007/s00013-005-1554-0
We give necessary and sufficient structural conditions on the ring B(R) of row and column finite matrices of a ring R which are equivalent to R being, respectively, Quasi-Frobenius, left artinian, and left noetherian.

Aurora Olivieri, Á. del Río y Juan Jacobo Simón
The group of automorphisms of the rational group algebra of a finite metacyclic group
Communications in Algebra 34 (2006) 3543-3567. DOI: 10.1080/00927870600796136
We give a method to compute the group of automorphisms of the rational group algebra QG, for G a finite metacyclic group.

Á. del Río y Sudarshan K. Sehgal
Zassenhaus Conjecture (ZC1) on torsión units of integral group rings for some metabelian groups
Archiv der Mathematik 86 (2006) 392-397. DOI: 10.1007/s00013-005-1554-0
We prove the First Zassenhaus Conjecture (each augmentation 1 periodic unit of the group of units of the integral group ring of a finite group G is conjutate in QG of an element of G) for some metabelian groups.

Antonio Pita y Á. del Río
Presentation of the group of units of Z D₁₆^-
Proceedings of “Groups, Rings and Group Rings”, Ubatuba, Brazil, 2004
Ser. Ledt. Notes in Pure and Appl. Math. Ed. A. Giambruno, C. Polcino Milies and S.K. Sehgal, Taylor and Francis Group, 2006, 305-314.
We compute a presentation of the group of units of ZD₁₆^-, where D₁₆^-=<a,b|a^8=b^2=1,ba=a^3b>.

Jeremy Haefner y Á. del Río
The Globalization Problem for inner automorphisms and Skolem-Noether Theorems
Proceedings of International Conference on Algebras, Modules and Rings, Lisbon 2003.
Ed. A. Facchini, K. Fuller, C.M. Ringel and C. Santa-Clara, World Scientific (2006), 37-51. DOI: 10.1142/9789812774552_0005
The Globalization Problem for inner automorphism of a ring with local units R asks whether given a familly of isomorphisms among the unital subrings of R, which are inner in a certain sense, there is a unique inner automorphsism of R which realize the isomorphisms of the familly by restriction. We show that, under certain conditions on R, the Globalization Problem for inner automorphism has a positive solution. As an application one obtain some Skolem-Noether like theorems for some infinite dimensional algebras. A counterexample for the Globalization Problem is also obtained.

Antonio Pita, Á. del Río y Manuel Ruiz
Groups of units of integral group rings of Kleinian type
Transactions of the American Mathematical Society 357 (8) (2005), 3215-3237. DOI: 10.1090/S0002-9939-05-08090-1
We introduce the notion of finite group of Kleinian type and classify the nilpotent finite groups of Kleinian type with nilpotency class 2. If G is a a finite group of Kleinian type then Poincaré's Method for the computation of presentations of Kleinian groups applies to compute presentations of a group commensurable with the group of units of ZG. However the Poincaré's Method is usually difficult to apply. We use Poincaré's Method to compute a presentation of the group of units of ZG for two finite groups G of order 16.

Capi Corrales, Eric Jespers, Guilherme Leal y Á. del Río
Presentation of the unit group of an order in a non-split quaternion algebra
Advances in Mathematics, 186 (2004) 498-524. DOI: 10.1016/j.aim.2003.07.015
We compute a presentation of the group of units of the quaternion ring H(O) = O + O i + O j + O k, where O is the ring of integers of K=Q(\sqrt{-7}) and 1,i,j,k is the canonical basis of the quaternion algebra H(K) (i^2=j^2=-1, k=ij=-ji). We also compute the cokernel of the cannonical homomorphism H(O)^*--->SO₃(O). Originally we wanted to compute the group of units of H(Z(e^2\pi
i/7), because this group is commensurable with the group of units of Z(Q₈\times C₇). The interest of this group is that the known methods to compute generators of a subgroup of finite index in the group of units of integral group rings do not apply to this group.

Aurora Olivieri, Á. del Río y Juan Jacobo Simón
On monomial characters and central idempotents of rational group algebras
Communications in Algebra, 32 (2004), no. 4, 1531-1550. DOI: 10.1081/AGB-120028797
Inspired by an idea of Jespers, Leal y Paques we show how to compute the primitive central idempotents of the rational group algebra QG, for G a finite monomial group and the Wedderburn decomposition of QG for G an abelian-by-supersolvable finite group.

Aurora Olivieri y Á. del Río
An algorithm to compute the primitive central idempotents and the Wedderburn decomposition of a rational group algebra
Journal of Symbolic Computations, 35 (2003) 673-687. DOI: 10.1016/S0747-7171(03)00035-X.
We present an effective algorithm to compute the Wedderburn decomposition of a rational group algebra or many finite groups, including all the abelian-by-supersolvable finite groups. This algorithm provides the first version of the GAP package wedderga.

Aurora Olivieri y Á. del Río
Bicyclic units of ZSn
Proceedings of the American Math. Soc. 131, (2003) 1649-1653. DOI: 10.1090/S0002-9939-03-06839-4
We show that the group generated by the bicyclic units of the symmetric group on four letters intersects the group of trivial units in the Klein group, generated by the products of two disjoint transpositions. This answer negatively Probelm 19 of S.K. Sehgal, Units in integral group rings, Longman Scientific and Technical Essex, 1993 which asked whether the group generated by the bicyclic units of a finite group is torsion-free.

Eli Aljadeff, Yuval Ginosar y Á. del Río
Semisimple Strongly Graded Rings
Journal of Algebra 256 (2002) 111-125. DOI: 10.1016/S0021-8693(02)00113-8
We give necessary and sufficient conditions for the primary objects mentioned in the previous paper give rise to a semisimple strongly graded ring. We also obtain a positive answer for the Twisting Problem for crossed products of cyclic groups. The Twisting Problem asks whether given an outer action of a group on a ring, there is a 2-cocycle, with respect to the given action, such that the crossed product obtained with the action and cocycle is semisimple.

Eric Jespers, Á. del Río y Manuel Ruiz
Groups generated by two bicyclic units in integral group rings
J. Group Theory 5(4) (2002) 493-511. DOI: 10.1515/jgth.2002.018, 17/09/2002
We show that if u an v are bicyclic units of the dihedral group of order 2p, with p prime, then the group generated by u an v is either free abelian or free non-abelian. Under a mild condition on u and v the hipothesis on the order of the group can be dropped.

Á. del Río y Manuel Ruiz
Computing large direct products of free groups in integral group rings
Communications in Algebra 30(4) (2002) 1751-1767. DOI: 10.1081/AGB-120013213
For each of the finite groups G classified in the previous paper, we compute a subgroup of minimal index in the group of units of the integral group ring ZG wich is a direct product of free groups.

Ernst Kleinert y Á. del Río
On the indecomposibility of unit groups
Abhandlungen Math. Sem. Hamburger 71 (2001) 291-295. DOI: 10.1007/BF02941478
We show that if G is the group of units of reduce norm 1 in a central simple algebra over a number field then G is virtually indecomposable as a direct product and as an amalgamated free product, except in the obvious cases where this cannot happen.

Á. del Río y Juan Jacobo Simón
Intermediate rings between matrix rings and Ornstein dual pairs
Archiv der Mathematiche, 75 (2000) 256-263. DOI: 10.1007/s000130050501
With the notation of the previous paper. Let A be a subring of E(R) containing B(R). We show that if A is Morita equivalent to E(S) (respectively, to B(S)) for some ring S, then A=E(R) (respectively, A=B(R)). More general results of this kind are obtained for matrix rings associated to Ornstein dual pairs.

Eric Jespers y Á. del Río
A structure theorem for the unit group of the integral group ring of some finite groups
Journal für die Reine und Angewandte Mathematik 521 (2000) 99-117. DOI: 10.1515/crll.2000.032, 02/05/2000
We clasify the finite nilpotent groups G such that the group of units of the integral group ring ZG is virtually a direct product of free groups.

Gene Abrams, Jeremy Haefner y Á. del Río
The Isomorphism Problem for Incidence Rings
Pacific Journal of Mathematics, 187 (1999) 201-214
Corrections and addenda to "The Isomorphism Problem for Incidence Rings"
Pacific Journal of Mathematics, 207(2) (2002) 497-506
The Isomorphism Problem for incidence rings asks whether two preordered sets P and Q are isomorphic if the incidence rings I(P,R) and I(Q,R), with coefficients in a given ring, are isomorphic.We show that if the ring R satify some finiteness conditions then the Isomorphism Problem has a positive solution for incidence rings of finite preordered sets with coefficients in R. This includes the case of R being noetherian, answering in the positive a question of Dascalescu and Van Wyk. On the oher side we show that given a family X of preordered sets, there is a ring R, such that R and I(P,R) are isomorphic for each P in X.

Gene Abrams, Jeremy Haefner y Á. del Río
Approximating rings with local units via automorphisms
Acta Math. Hungar. 82 (1999), no. 3, 229-248. DOI: 10.1023/A:1026460815618
We show that if A is a ring of local units then every inner automorphism of A is the restriction of an inner automorphism of the ring of multipliers of A. We also show that this is no a similar natural overring which satisfy the corresponding property for all the automorphisms or for the outer automorphisms.

Jeremy Haefner y Á. del Río
Actions of Picard groups on graded rings
Journal of Algebra 218 (1999) 573-607. DOI: 10.1006/jabr.1999.7862
We introduce an action of the Picard group of a ring A on the graded rings for which A is isomorphic to R₁, the subring of homomgeneous elements of degree 1. We show that the orbit of a strongly graded ring R is the formed by the strongly graded rings S for which ther is a graded equivanlence between R-gr and S-gr. As an application one construct, from primary objects, a familly of strongly graded rings containing all the semisimple strongly graded rings.

Margaret Beattie y Á. del Río
Graded equivalences and Picard groups
Journal of Pure and Applied Algebra. 141 (1999), no. 2, 131-152. DOI: 10.1016/S0022-4049(98)00011-5
We study the Picard group of the category of graded modules of a group graded ring and some of its subgroups.

Jeremy Haefner, Á. del Río y Juan Jacobo Simón
Isomorphisms of row and column finite matrix rings
Proceeding of the American Math. Soc. 125 (1997), 1651-1658. DOI: 10.1090/S0002-9939-97-03849-5
Let R be a unital ring. Let I be the ring of matrices indexed by an infinite set with entries in R having finitely many non-zero entries, and B=B(R) the ring of matrices with row and finite columns. We show that I is fixed by each automorphism of R. As an application we show that two unital rings R and S are Morita equivalent if and only if B(R) and B(S) are isomorphic. We also show that, while the ring E(R) of finite columns of R is not Morita equivalent to B(S) for every ring S, the Picard groups of E(R) and B(R) are isomorphic.

Guilherme Leal y Á. del Río
Products of Free Groups in the Unit Group of Integral Group Rings II
Journal of Algebra 191 (1997), 240-251. DOI: 10.1023/A:1006675731141
We clasify the finite groups G such that the group of units of the integral group ring ZG is virtually a direct product of non-abelian free groups.

Eric Jespers, Guilherme Leal y Á. del Río
Products of Free Groups in the Unit Group of Integral Group Rings
Journal of Algebra 180 (1996) 22-40. DOI: 10.1006/jabr.1996.0050
We clasify the finite nilpotent groups G such that the group of units of the integral group ring ZG is virtually a direct product of non-abelian free groups.

Edgar E. Enochs, Juan José García y Á. del Río
When does R Gorenstein does implies R^G Gorenstein?
Journal of Algebra 182 (1996) 561-576. DOI: 10.1006/jabr.1996.0190
We show that under certain conditions, if a ring R is Gorenstein, then so is the ring of invariants R^G under the action of a group G on R.

Michael Clase, Eric Jespers y Á. del Río
Semigroup Graded Rings with finite support
Glasgow Mathematics Journal, 38 (1996) 11-18. DOI: 10.1017/S0017089500031190
We show that if R is an S-graded ring with finite support, for S a semigroup, then S is perfect if and only if so is the subring R_e of homogeneous elements of degree e, for every idempotent e of R. Other results of this kind are obtained.

Margaret Beattie y Á. del Río
The Picard group of a category of graded modules
Communications in Algebra 24 (1996), 4397-4414. DOI: 10.1080/00927879608825823
Se estudia el grupo de Picard de la categoría de módulos graduados de un anillo graduado y algunos subgrupos suyos.

Sorin Dascalescu, C. Nastasescu, Á. del Río y Fred Van Oystaeyen
Gradings of Finite Support. Applications to Injective Objects
Journal of Pure and Applied Algebra, 107 (1996) 193-206. DOI: 10.1016/0022-4049(95)00063-1
We show that if M is a group graded module with finite support then M is injective if and only if it is gr-inyective.

Juan José García y Á. del Río
On Flatness and Projectivity of a Rings as a Module over a Fixed Subring
Mathematica Scandinavica, 76 (1995) 179-193.
We study necessary and sufficient conditions for a ring R to be projective as a module over the ring of invariants R^G under the action of a group G on R.

Ricardo Alfaro, Pere Ara y Á. del Río
Regular Skew Group Rings
Journal of Australian Mathematical Society (Series A), 54 (1995), 167-182
We study necessary conditions and sufficient conditions for a skew group ring to be Von Neummann regular.

Juan José García y Angel del Río
Actions of Groups on Fully Bounded Noetherian Rings
Communications in Algebra 22 (1994) 1495-1505. DOI: 10.1080/00927878908823866
We show that under certain hipothesis, if the ring of invariants R^G of an action of a group G on a ring R is FBN, then R is also FBN. This answer in the positive a question of Fischer and Osterburg.

Gene Abrams, Claudia Menini y Angel del Río
Realization Theorems for Categories of Graded modules over Semigroup Graded Rings
Communications in Algebra 22 (1994) 5343-5388. DOI: 10.1080/00927879408825135
We study functors between categories of modules graded by semigroups.

Á. del Río
On Quasi-Frobenius Pure Semisimple Rings
Bulletin de la Societe Mathematique de Belgique 45 (1993) 117-121
We show that the conjecture on pure-semisimple rings for quasi-Frobenius rings is equivalente to a topology-algebraic connection.

Sorín Dascalescu y Á. del Río
Graded T-Rings with Finite Support
Communications in Algebra 21 (1993) 3619-3636. DOI: 10.1080/00927879308824752
We show that the a group graded ring with finite support is a T-ring (also knonw as an FBN ring) if and only if the subring of homogeneous elements of degree 1 is a T-ring.

Á. del Río
Categorical Methods in Graded Ring Theory
Publications Matemàtiques 36 (1992) 489-531. DOI: 10.5565/PUBLMAT_362A92_15
This paper is a survey on categorical methods in graded ring theory. The results of the previous papers are generalize to categories of modules graded by G-sets and one obtain several applications.

Claudia Menini y Á. del Río
Morita Dualities and Graded Rings
Communications in Algebra 19, (1991) 1765-1794. DOI: 10.1080/00927879108824228
We give versions of the Morita Theorems for dualities between categories of group graded modules. Some applications are provided.

Á. del Río
Graded Rings and Equivalenes of Categories
Communications in Algebra 19, (1991) 997-1012. DOI: 10.1080/00927879108824184
Corrections
Communications in Algebra 23 (1995) 3943-3946. DOI: 10.1080/00927879508825440
We give versions of the Morita Theorems for equivalencies between categories of group graded modules. Some applications are provided.

Á. del Río y Manuel Saorín
Dualities and dimensions of Endomorphism Rings
Tsukuba Journal of Mathematics 15, (1991) 1-18
We show how to compute the global and weak dimensions of a quasi-injective module using duality techniques for topological modules.

Á. del Río
Weak Dimension of Group Graded Rings
Publicacions Matemàtiques 34 (1990) 209-216. DOI: 10.5565/PUBLMAT_34190_16
We prove that the weak dimension of a G-graded ring R coincides with the graded weak dimension under the hipothesis that G is locally finite and the adjoint funtor of the forgetful functor of R_H is separable for each finite subgroup H of G.

M.D. Rafael (M. Saorín, D. Herbera, R. Colpi, Á. del Río, F. Van Oystaeyen, A. Giaquinto, E. Gregorio y L. Biondi)
Separable Functors Revisited
Communications in Algebra 18, (1990) 1445-1459. DOI: 10.1080/00927879008823975
We characterize the separable adjoint functors in terms of the unit and counit. The authors are the participants in a summer couse held in Cortona (Italy) given by F. Van Oystaeyen.

Á. del Río
Bigraded Bimodules
Proceedings of the Spanish-Belgian Week on Algebra and Geometry, F. Gago and E. Villanueva Editors, (1989) 167-176
We introduce the notion of bigraded bimodule and show that the adjoint functors between categories of graded modules of group graded rings can be realize as Hom and tensor funtors of bigraded bimodules.

Pere Ara y Á. del Río
A Question of Passman on the Symmetric Ring of Quotients
Israel Journal of Mathematics 68 (1989) 348-352. DOI: 10.1007/BF02764989
We obtain an example of a prime ring R such that the sequence (R_n) of Martindale symmetric ring of quotients (R₀=R, R_n+1=Q(R_n)) does not stabilize.

Á. del Río y Manuel Saorín
Dualities and Lattice Isomorphisms
Actas XIII Jornadas Hispano-Lusas de Matemática, Valladolid
We show a Galois connection between the lattices of submodules associated by a duality between categories of topological modules.

Á. del Río
Self-Injective Endomorphism Rings of Quasi-Injective Modules
Communications in Algebra 17, (1989) 2611-2634. DOI: 10.1080/00927878908823866
We characterize the self-injective endomorphism rings of quasi-injective modules in terms of properties of the module. We use techniques of dualities between categories topological modules.

Á. del Río
Condiciones de Finitud en Anillos de Endomorfismos de Módulos Quasi-Inyectivos
Actas XII Jornadas Hispano-Lusas de Matemáticas, Vol. 2, (1987) 147-152
We show that some finiteness conditions on the endomorphism ring of a quasi-injective module can be expressed in terms of properties of the module.