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Research
My papers
Main research interest: Units of group rings and orders, group algebras and algebraic codes.
I also contributed on some other topics of Ring and Group Theory out of the main topics of interest.
The papers are listed in decreasing order of appearance.
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D. García Lucas, Á. del Río
A reduction theorem for the Isomorphism Problem of group algebras over fields
Journal of Pure and Applied Algebra, 228 no. 4, Paper no. 107511, 6 pp.
doi.org/10.1016/j.jpaa.2023.107511: Open Access.
We prove that the Isomorphism Problem for group algebras reduces to group algebras over finite extensions of the prime field. In particular, the Modular Isomorphism Problem reduces to finite modular group algebras.
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O. Broche, D. García Lucas, Á. del Río
A classification of the finite 2-generator cyclic-by-abelian groups of prime-power order
International Journal of Algebra and Computation, 33 no. 04 (2023) 641-686.
dx.doi.org/10.1142/S0218196723500297
We classify the finite 2-generator cyclic-by-abelian groups of prime-power order. We associate to each such group $G$ a list inv(G) of numerical group invariants which determines the isomorphism type of G. Then we describe the set formed by all the possible values of inv(G).
This allows us to develop practical algorithms to construct all finite non-abelian 2-generator cyclic-by-abelian groups of a given prime-power order, to compute the invariants of such a group, and to decide whether two such groups are isomorphic.
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À. García-Blázquez, Á. del Río
A classification of metacyclic groups by group invariants
Bull. Math. Soc. Sci. Math. Roumanie Tome 66 (114), No. 2, 2023, 209-233.
We obtain a new classification of the finite metacyclic group in terms of group invariants. We present an algorithm to compute these invariants, and hence to decide if two given finite metacyclic groups are isomorphic, and another algorithm which computes all the metacyclic groups of a given order. A GAP implementation of these algorithms is given.
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S. Chagas, Á. del Río, P. Zalesskii
Aritmethic lattices of SO(1,n) and units of group rings
Journal of Pure and Applied Algebra, 227 (2023), no. 11, Paper No. 107405, 17 pp.
doi.org/10.1016/j.jpaa.2023.107405
We establish that standard arithmetic subgroups of a special orthogonal group SO(1,n) are conjugacy separable. As an application we deduce this property for unit groups of certain integer group rings.
We also prove that finite quotients of group of units of any of these group rings determines the original group ring.
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D. García-Lucas, Á. del Río, M. Stanojkovski
On group invariants determined by modular group algebras: Even versus odd characteristic
Algebras and Representation Theory (2022).
doi.org/10.1007/s10468-022-10182-x: Open Access
Let p be a an odd prime and let G be a finite p-group with cyclic commutator subgroup G'. We prove that the exponent and the abelianization of the centralizer of G' in G are determined by the group algebra of G over any field of characteristic p. If, additionally, G is 2-generated then almost all the numerical invariants determining G up to isomorphism are determined by the same group algebras; as a consequence the isomorphism type of the centralizer of G' is determined. These claims are known to be false for p = 2.
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A. Bächle, A. Kiefer, S. Maheshwary, Á. del Río
Gruenberg-Kegel graphs: cut groups, rational groups and the Prime Graph Question
Forum Mathematicum, 35 (2023) nº2 409-429.
doi.org/10.1515/forum-2022-0086
The Gruenberg-Kegel graph of a group is the undirected graph whose vertices are those primes which occur as the order of an element of the group, and distinct vertices p, q are joined by an edge whenever the group has an element of order pq.
A group is said to be cut if the central units of its integral group ring are trivial. We give a complete classification of the Gruenberg-Kegel graphs of finite solvable cut groups which have at most three elements in their prime spectrum. For the remaining cases of finite solvable cut groups, we strongly restrict the list of the possible Gruenberg-Kegel graphs and realize most of them by finite solvable cut groups. Likewise, we give a list of the possible Gruenberg-Kegel graphs of finite solvable rational groups and realize as such all but one of them. As an application, we completely classify the Gruenberg-Kegel graphs of metacyclic, metabelian, supersolvable, metanilpotent and 2-Frobenius groups for the classes of cut groups and rational groups, respectively. The Prime Graph Question asks whether the Gruenberg-Kegel graph of a group coincides with that of the group of normalized units of its integral group ring. We answer the Prime Graph Question for integral group rings for finite rational groups and most finite cut groups.
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D. García-Lucas, L. Margolis, Á. del Río
Non-isomorphic 2-groups with isomorphic modular group algebras
Journal fur die Reine und Angewandte Mathematik, 783 (2022) 269-274.
doi.org/10.1515/crelle-2021-0074
We provide non-isomorphic finite 2-groups which have isomorphic group algebras over any field of characteristic 2, thus settling the Modular Isomorphism Problem.
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O. Broche, Á. del Río
The Modular Isomorphism Problem for two generated groups of class two
Indian Journal of Pure and Applied Mathematics, 52 (2021) 721-728.
doi.org/10.1007/s13226-021-00182-w
We prove that if G is a finite 2-generated $p$-group of nilpotence class at most 2 then the group algebra of G with coefficients in the field with $p$ elements determines $G$ up to isomorphisms.
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M. Caicedo, Á. del Río
On the Zassenhaus Conjecture for certain cyclic-by-nilpotent groups
Mediterranean Journal of Mathematics, 17, 62 (2020).
doi.org/10.1007/s00009-020-1479-7
We study the Zassenhaus Conjecture for the class of cyclic-by-nilpotent groups with special attention to the class of cyclic-by-Hamiltonian groups.
We prove the conjecture for cyclic-by-$p$-groups and some cyclic-by-Hamiltonian groups.
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L. Margolis, Á. del Río, M. Serrano
Zassenhaus Conjecture on torsion units holds for PSL(2,p) with p a Fermat or Mersenne prime
Advances in Group Theory and Applications, 8 (2019), 1–37
doi.org/10.32037/agta-2019-009
We prove the Zassenhaus Conjecture for the groups $\PSL(2,p)$, where $p$ is a Fermat or Mersenne prime. Our result is an easy consequence of known results and our main theorem which states that the Zassenhaus Conjecture holds for a unit in $\Z\PSL(2,q)$ of order coprime with $2q$, for some prime power $q$.
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L. Margolis, Á. del Río
Finite Subgroups of Group Rings: A survey
Advances in Group Theory and Applications, 8 (2019), 1–37
doi.org/10.32037/agta-2019-009
In this survey we revise the state of the art on the study of the finite subgroups of group rings, including classical and recent results and sketching some of the methods used in the study of the main problems as for example the Isomorphism and Automorphism Problems, the Zassenhaus Conjectures and the Prime Graph Question.
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Á. del Río, M. Serrano
Zassenhaus conjecture on torsion units holds for SL(2,p) and SL(2,p^2)
Journal of Group Theory 22 (2019), 953–974.
doi:10.1016/10.1515/jgth-2018-0113
We prove the Zassenhaus conjecture for the groups SL(2,p) and SL(2,p^2) with p a prime number. We also prove that if G=SL(2,q), with q an arbitrary prime power and u is a torsion unit of ZG with augmentation 1 and order coprime with p, then u is conjugate in QG to an element of G. By known results, this reduces the proof of the Zassenhaus conjecture for these groups to proving that every unit of ZG of order a multiple of p and augmentation 1 has order actually equal to p.
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L. Margolis, Á. del Río
Partial augmentations power property: A Zassenhaus conjecture related problem
Journal of Pure and Applied Algebra 223 (2019) 4081-4101.
doi:10.1016/j.jpaa.2018.12.018
We review the known weaker versions of this conjecture and introduce a new condition, on the partial augmentations of the powers of a unit of finite order in $\mathbb{Z}G$, which is weaker than the Zassenhaus Conjecture but stronger than its other weaker versions.
We prove that this condition is satisfied for units mapping to the identity modulo a nilpotent normal subgroup of $G$. Moreover, we show that if the condition holds then the HeLP Method adopts a more friendly form and use this to prove the Zassenhaus Conjecture for a special class of groups.
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O. Broche, J.Z. Gonçalves, Á. del Río
Group algebras whose units satisfy a Laurent polynomial identity
Archiv der Mathematik 111 (4) (2018), 353-367.
doi:10.1007/s00013-018-1223-8
Let KG be the group algebra of a torsion group G over a field K. We show that if the units of KG satisfy a
Laurent polynomial identity, which is not satisfied by the units of the relative free algebra K[alpha,beta :
alpha^2=beta^2=0] then KG satisfies a polynomial identity. This extends Hartley's Conjecture which states that if
the units of KG satisfy a group identity, then KG satisfies a polynomial identity. As an application we prove that
if the units of KG satisfy a Laurent polynomial identity whose support has cardinality at most 3, then KG
satisfies a polynomial identity.
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L. Margolis, Á. del Río
An algorithm to construct candidates to counterexamples to the Zassenhaus Conjecture
Journal of Algebra 514 (2018) 536-558.
doi:10.1016/j.jalgebra.2018.06.026
Let G be a finite group, N a nilpotent normal subgroup of G and let V(ZG, N) denote the group formed by the units of the integral group ring ZG of G which map to the identity under the natural homomorphism ZG \rightarrow Z(G/N).
Sehgal asked whether any torsion element of V(ZG, N) is conjugate in the rational group algebra of G to an element of G.
This is a special case of the Zassenhaus Conjecture.
By results of Cliff and Weiss and Hertweck, Sehgal's Problem has a positive solution if N has at most one non-cyclic Sylow subgroup.
We present some algorithms to study Sehgal's Problem when N has at most one non-abelian Sylow subgroup.
They are based on the Cliff-Weiss inequalities introduced by the authors in a paper mentioned below.
With the help of these algorithms we obtain some positive answers to Sehgal's Problem and use them to show that for units in
V(ZG,N) our method is strictly stronger than the well known HeLP Method.
We then present a method to use the output of one of the algorithms to construct explicit metabelian groups which are candidates to a negative solution to Sehgal's Problem.
Recently Eisele and Margolis showed that some of the examples proposed in this paper are indeed counterexamples to the Zassenhaus Conjecture. These are the first known counterexamples. Moreover, we prove that every metabelian negative solution of Sehgal's Problem satisfying some minimal conditions is given by our construction.
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L. Margolis, Á. del Río
Cliff-Weiss Inequalities and the Zassenhaus Conjecture
Journal of Algebra, 507 (2018) 292-319.
doi:10.1016/j.jalgebra.2018.04.019
Let N be a nilpotent normal subgroup of the finite group G.
Assume that u is a unit of finite order in the integral group ring ZG of G which maps to the identity in Z(G/H). We show how a result of Cliff and Weiss can be used to derive linear inequalities on the partial augmentations of u and apply this to the study if u is conjugate to an element of G within the rational group algebra QG.
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Á. del Río, M. Serrano
On the torsion units of the integral group ring of finite projective special linear groups
Communications in Algebra, 45:12 (2017), 5073-5087.
doi:10.1080/00927872.2017.1291814
H. J. Zassenhaus conjectured that any unit of finite order and augmentation one in the integral group ring of a finite group G is conjugate in the
rational group algebra to an element of G. One way to verify this is showing that such unit has the same distribution of partial augmentations
as an element of G and the HeLP Method provides a tool to do that in some cases. In this paper we use the HeLP Method to describe the partial
augmentations of a hypothetical counterexample to the conjecture for the projective special linear groups.
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Á. del Río, P. Zalesskii
Coherent groups of units of integral group rings and direct products of free groups
Mathematical Proceedings of the Cambridge Philosophical Society, 162(2) (2017), 191-209.
doi: 10.1017/S0305004116000517.
We classify the finite groups G for which U(ZG), the group of units of the integral group ring of G,
does not contain a direct product of two non-abelian free groups.
These list of groups contains all the groups for which U(ZG) is coherent.
This reduces the problem to classify the finite groups G for which U(ZG) is coherent to decide about the
coherency of a finite list of groups of the form SL(n,R), with R an order in a finite dimensional rational
division algebra.
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O. Broche, Á. del Río,
Polynomials of degree 4 defining units
Rev. Mat. Iberoam. 33 (2017) 1487-1499.
doi: 10.4171/RMI/979
We obtain a general form for all the polynomials of degree 4 witin integral coefficients which provides a
units of the integral group ring when evaluated on a group element of finite order.
We obtain a full classification of such polynomials when the group element has order at most 10 and
we provide a strategy to obtain similar results for larger order.
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O. Broche, Á. del Río,
Polynomials defining many units
Math. Z. 283 (2016), 3-4, pp 119-1200,
doi: 10.1007/s00209-016-1638-5
We classify the polynomials with integral coefficients that, when evaluated on a group element of finite order n, define a unit in the integral group ring for infinitely many positive integers n.
We show that this happens if and only if the polynomial defines generic units in the sense of Marciniak and Sehgal.
We also classify the polynomials with integral coefficients which provides units when evaluated on n-roots of a fixed integer a for infinitely many positive integers n.
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E. Jespers, A. Keifer, Á. del Río,
Presentations of Groups Acting Discontinuously on Direct Products of Hyperbolic Spaces
Math. Comp. 85 (2016), 2515-2552.
doi:10.1090/mcom/3071
We investigate a extension of the presentation part of Poincaré Theorem to some discontinuous groups of isometries of hyperbolic spaces.
As a study case we consider a quaternion algebra A over a number field and using the different representations of A as 2-by-2 matrices over the complex numbers we consider the group G of reduced norm elements of an order in A acting discontinuously on a direct product of copies of 2-dimensional and 3-dimensional hyperbolic spaces.
We initiate this approach by executing this method on the Hilbert modular group, i.e. the projective linear group of degree two over the ring of integers of a real quadratic extension of the rationals. This group acts discontinuously on a direct product of two hyperbolic spaces of dimension two.
We construct a fundamental domain analogue to the Ford domain of a Fuchsian or a Kleinian group and find a method to obtain a presentation of the Hilbert modular group of the same type as the presentation given by Poincaré Theorem in the case of Kleinian groups.
For the applications we have in mind the groups of units of integral group rings. This first case give some hope that one can obtain a more general result for arbitrary quaternion algebras over number fields and then apply this to obtain presentations of groups of units of some integral group rings, extending the method introduced in [Pita, del Río, Ruiz, Groups of units of integral group rings of Kleinian type, Transactions of the American Mathematical Society 357 (8) (2005), 3215-3237.]
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E. Jespers, A. Keifer, Á. del Río,
Revisiting Poincaré's Theorem on presentations of discontinuous groups via fundamental polyhedra
Expositiones Mathematicae 33 (2015) 401-430.
doi:10.1016/j.exmath.2015.01.001
We give a new proof of the Poincaré Theorem which states that the pairing transformations together with the reflexion and cycle relations,
fundamental polihedron of a discontinuous group of isometries of a Riemman variety of constant curvaturs, form a presentation of the group.
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O. Broche Cristo, Á. del Río, M. Ruiz
Group rings whose set of symmetric elements is Lie metabelian
Forum Mathematicum 27(6) (2015) 3533-3566.
doi: 10.1515/forum-2013-0181
We classify the group rings of zero characteristic for which the symmetric elements satisfy the identity [[x,y],[z,w]]=0.
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M.J. Caicedo, Á. del Río
On the Congruence Subgroup Problem for integral group rings
Journal of Algebra
doi: 10.1016/j.jalgebra.2014.01.029
We give a short list of finite groups satisfying the following property:
If G is a finite group such that the Congruence Subgroup Problem has a negative solution for the group of units U(ZG) of the integral group ring of G
then one of the groups in the list is an epimorphic images of G.
Moreover, for many of the groups G in the list the Congruence Subgroup Problem has a negative solution for U(ZG) and hence if G is an epimorphic image
of a group H then the Congruence Subgroup Problem has a negative solution for H.
The list is minimal in the following sense: If H is a proper epimorphic image of G then the Congruence Subgroup Problem has a positive solution for U(ZG).
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J.Z. Gonçalves, R.M. Guralnick, Á. del Río
Bass units as free factors in integral group rings of simple groups
Journal of Algebra, 404 (2014) 100?123.
doi: 10.1016/j.jalgebra.2013.12.024
We first classify the simple groups with a dihedral p-critical element.
They are all of the form PSL(2,q).
We then extends the main result of "Bass cyclic units as factors in a free group in integral group ring units" to the group of this form with q a power of 2
and for some cases with q an odd prime power. This gives support to the following conjecture:
Let G be a finite group, u a Bass unit based on an element a of G of prime order, and assume that u has infinite order modulo the centre of U(ZG).
Then there is a Bass unit or a bicyclic unit v and a positive integer n such that the group generated by u^n and v^n is a
non-abelian free group.
To prove the conjecture it remains to prove it for PSL(2,q) with q an odd prime power.
With results of the paper and the help of computers we verify the conjectture for all q<10000.
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Eric Jespers, Gabriela Olteanu, Á. del Río, Inneke Van Gelder
Group rings of finite strongly monomial groups: central units and primitive idempotents
Journal of Algebra. 387 (2013) 99?116.
doi: 10.1016/j.jalgebra.2013.04.020
We compute the rank of the group of central units in the integral group ring ZG of a finite strongly monomial group G.
The formula obtained is in terms of the strong Shoda pairs of G.
Next we construct a virtual basis of the group of central units of ZG for a class of groups G properly contained in the finite strongly monomial groups.
Furthermore, for another class of groups G inside the finite strongly monomial groups, we give an explicit construction of a complete set of orthogonal primitive idempotents of QG.
Finally, we apply these results to describe finitely many generators of a subgroup of finite index in the group of units of ZG,
this for metacyclic groups G which are the semidirect product of a cyclic q-group by a cyclic p-group with p and q different primes and the cyclic
p-group acting faithfully on the cyclic q-group.
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Josep Rifà, Á. del Río
Families of Hadamard Z_2Z_4Q_8-codes
IEEE Transactions of Information Theory, 59 (2013), no. 9 5140-5151.
doi: 10.1109/TIT.2013.2258373
10.1109/TIT.2013.2258373
A Z_2Z_4Q_8-code is the binary image, after a Gray map, of a non-empty subgroup of a direct product of copies of the cyclic groups of order 2
and 4 and the quaternion group of order 8.
Such Z_2Z_4Q_8-codes are translation invariant propelinear codes as are the well known Z_4-linear or Z_2Z_4-codes.
In the current paper, we show that there exist ``pure'' Z_2Z_4Q_8-codes, that is, codes that do not admit any abelian translation invariant
propelinear structure.
We study the dimension of the kernel and rank of the Z_2Z_4Q_8-codes, and we give upper and lower bounds for these parameters.
We give tools to construct a new class of Hadamard codes formed by several families of Z_2Z_4Q_8-codes;
we classify such codes from an algebraic point of view and we improve the upper and lower bounds for the rank and the dimension of the kernel
when the codes are Hadamard.
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Mauricio Caicedo, Leo Margolis, Á. del Río,
Zassenhaus conjecture for cyclic-by-abelian groups
J. London Math. Soc. (2) 88 (2013) 65-78.
doi: 10.1112/jlms/jdt002
Zassenhaus Conjecture for torsion units states that every augmentation one torsion unit of the integral group
ring of a finite group G is conjugate to an element of G in the units of the rational group algebra Q G.
This conjecture has been proved by Weiss for nilpotent groups, by Hertweck by some families of metabelian groups
including all metacyclic groups and by several other authors for some other families of groups.
We prove the conjecture for cyclic-by-abelian groups.
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Jairo Z. Gonçalves, Á. del Río,
A survey on free subgroups in the group of units of group rings
J. Algebra Appl. 12 (2013), no. 6, 1350004, 28 pp
doi: 10.1142/S0219498813500047
In this survey we revise the methods and results on the existence and construction of free groups of units in
group rings, with special emphasis in integral group rings over finite groups and group algebras.
We also survey results on constructions of free groups generated by elements which are either symmetric or
unitary with respect to some involution and other results on which integral group rings have large subgroups
which can be constructed with free subgroups and natural group operations.
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Eric Jespers, Gabriela Olteanu, Á. del Río, Inneke Van Gelder
Central units of integral group rings
Proc. Amer. Math. Soc. 142 (2014), no. 7, 2193-2209.doi: 10.1090/S0002-9939-2014-11958-7
We give an explicit description for a basis of a subgroup of finite index in the group of central units of the integral group ring
ZG of a finite abelian-by-supersolvable group such that every cyclic subgroup of order not a divisor of 4 or 6 is subnormal in G. The basis elements turn out to
be a natural product of conjugates of Bass units. This extends and generalizes a result of Jespers, Parmenter and Sehgal showing that the Bass units
generate a subgroup of finite index in the center of U(ZG) of the unit group U(ZG) in case G is a finite nilpotent group.
Next, we give a new construction of units that generate a subgroup of finite index in the center of U(ZG) for all finite strongly monomial groups
G. We call these units generalized Bass units. Finally, we show that the commutator group U(ZG)/U(ZG)' and the center U(ZG) have the
same rank if G is a finite group such that QG has no epimorphic image which is either a non-commutative division algebra other than a totally
definite quaternion algebra, or a two-by-two matrix algebra over a division algebra with center either the rationals or a quadratic imaginary
extension of Q. This allows us to prove that in this case the natural images of the Bass units of ZG generate a subgroup of finite index in
U(ZG)/U(ZG).
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Eric Jespers, Á. del Río, I. Van Gelder
Writing units of integral group rings of finite abelian groups as a product of Bass units
Mathematics of Computation, Published electronically: May 2013.
doi: 10.1090/S0025-5718-2013-02718-4
We give a constructive proof of the theorem of Bass and Milnor saying that if G is
a finite abelian group then the Bass units of the integral group ring ZG generate a subgroup of
finite index in its units group U(ZG). Our proof provides algorithms to represent some units
that contribute to only one simple component of QG and generate a subgroup of finite index in
U(ZG) as product of Bass units. We also obtain a basis B formed by Bass units of a free abelian
subgroup of finite index in U(ZG) and give, for an arbitrary Bass unit b, an algorithm to express
b^{\varphi(|G|)} as a product of a trivial unit and powers of at most two units in this basis B.
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Á. del Río, Manuel Ruiz, Pavel Zalesskii
Subgroup separability in integral group rings
Journal of Algebra 347 (2011) 60?68.
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.
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Eric Jespers, Gabriela Olteanu, Á. del Río
Rational group algebras of finite groups:
from idempotents to units of integral group rings
Algebras and Representation Theory, 15 (2012) 359-377.
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 ZG 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.
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Jairo Z. Gonçalvez, 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.
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José Joaquín Bernal, Á. del Río, Juan Jacobo 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.
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Ferran Cedó, Eric Jespers, 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 and 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.
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Allen Herman, Gabriela Olteanu, 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.
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José Joaquín Bernal, Á. del Río, Juan Jacobo 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.
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José J. Bernal, Á. del Río, Juan Jacobo Simón
An intrinsical description of group codes
Designs, Codes and Cryptography 51, nº 3 (2009) 289-300.
doi: 10.1007/s10623-008-9261-z 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, 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, 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, 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, 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, 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.
-
Gurmeet K. Bakshi, Osnel Broche Cristo, Allen Herman, Alexander Konovalov, Sughanda Maheshwary, Aurora Olivieri, Gabriela Olteanu, Á. del Río, Inneke Van Gelder
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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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(e2pi
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, 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 and 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, Á. del Río
Products of Free Groups in the Unit Group of Integral Group Rings II
Journal of Algebra 191 (1997), 240-251.
doi: 10.1006/jabr.1996.0050
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, 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, Á. del Río
When does R Gorenstein does implies RG 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
RG under the action of a group G on R.
-
Michael Clase, Eric Jespers, 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 Re
of homogeneous elements of degree e, for every idempotent e of R.
Other results of this kind are obtained.
-
Margaret Beattie, Á. del Río
The Picard group of a category of graded modules
Communications in Algebra 24 (1996), 4397-4414.
doi: 10.1080/00927879608825823
We study the Picard group of a the category of graded modules
of a group graded ring and some of its subgroups.
-
Sorin Dascalescu, C. Nastasescu, Á. del Río, 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, 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
RG under the action of a group G on R.
-
Ricardo Alfaro, Pere Ara, 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, 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 RG 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, 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, 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, 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, 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 RH 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, 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, 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 (Rn)
of Martindale symmetric ring of quotients (R0=R,
Rn+1=Q(Rn)) does not stabilize.
- Á. del Río, 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.
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