Ángel Del Río

Ángel Del Río
University of Murcia | UM · Department of Mathematics

Ph.D. Science Mathematics

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112
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Publications (112)
Preprint
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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 $pq$. It reflects interesting properties of the group. A group is said to be cut if the central units of its...
Article
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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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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.
Preprint
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We provide non-isomorphic finite 2-groups which have isomorphic group algebras over any field of characteristic 2, thus settling the Modular Isomorphism Problem.
Preprint
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We obtain a classification of the finite two-generated cyclic-by-abelian groups of prime power order. For that 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$(G)$.
Preprint
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We prove that if $G$ is 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.
Article
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Hans Zassenhaus conjectured that every torsion unit of the integral group ring of a finite group G is conjugate within the rational group algebra to an element of the form \(\pm g\) with \(g\in G\). This conjecture has been disproved recently for metabelian groups, by Eisele and Margolis. However, it is known to be true for many classes of solvable...
Preprint
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Hans Zassenhaus conjectured that every torsion unit of the integral group ring of a finite group $G$ is conjugate within the rational group algebra to an element of the form $\pm g$ with $g\in G$. This conjecture has been disproved recently for metabelian groups, by Eisele and Margolis, but before it had been proved for many classes of solvable gro...
Preprint
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In the 1940's Graham Higman initiated the study of finite subgroups of the unit group of an integral group ring. Since then many fascinating aspects of this structure have been discovered. Major questions such as the Isomorphism Problem and the Zassenhaus Conjectures have been settled, leading to many new challenging problems. In this survey we rev...
Article
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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→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 spec...
Preprint
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We revise some problems on the study of finite subgroups of the group of units of integral group rings of finite groups and some techniques to attack them.
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H. J. Zassenhaus conjectured that any unit of finite order and augmentation 1 in the integral group ring {\mathbb{Z}G} of a finite group G is conjugate in the rational group algebra {\mathbb{Q}G} to an element of G . We prove the Zassenhaus conjecture for the groups {\mathrm{SL}(2,p)} and {\mathrm{SL}(2,p^{2})} with p a prime number. This is the fi...
Article
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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 Conjecture which states that if the uni...
Article
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If x is the generator of a cyclic group of order n then every element of the group ring ℤ{x} is the result of evaluating x at a polynomial of degree smaller than n with integral coefficients. When such an evaluation result into a unit we say that the polynomial defines a unit on order n. Marciniak and Sehgal have classified the polynomials of degre...
Article
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Let $G$ be a finite group, $N$ a nilpotent normal subgroup of $G$ and let $\mathrm{V}(\mathbb{Z} G, N)$ denote the group formed by the units of the integral group ring $\mathbb{Z} G$ of $G$ which map to the identity under the natural homomorphism $\mathbb{Z} G \rightarrow \mathbb{Z} (G/N)$. Sehgal asked whether any torsion element of $\mathrm{V}(\m...
Article
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Zassenhaus conjectured that any unit of finite order in the integral group ring $\mathbb{Z}G$ of a finite group $G$ is conjugate in the rational group algebra of $G$ to an element in $\pm G$. 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...
Article
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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 $\mathbb{Z} G$ of $G$ which maps to the identity under the linear extension of the natural homomorphism $G \rightarrow G/N$. We show how a result of Cliff and Weiss can be used to derive linear inequalities on the par...
Article
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H.J. Zassenhaus conjectured that any unit of finite order in the integral group ring $\mathbb{Z}G$ of a finite group $G$ is conjugate in the rational group algebra $\mathbb{Q}G$ to an element of the form $\pm g$ with $g \in G$. Though known for some series of solvable groups, the conjecture has been proved only for thirteen non-abelian simple group...
Article
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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 w...
Article
We classify the finite groups G for which \$\mathcal{U}({\mathbb Z} G)\$, the group of units of the integral group ring of G, does not contain a direct product of two non-abelian free groups. This list of groups contains all the groups for which \$\mathcal{U}({\mathbb Z} G)\$ is coherent. This reduces the problem to classify the finite groups G for...
Article
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Let $G$ be a finite group. Zassenhaus conjectured that every torsion unit of augmentation one of the integral group ring of $G$ is conjugate in the rational group algebra to an element of $G$. The HeLP Method provides a technique to prove this conjecture for an element $u$ of order $n$, by showing that the partial augmentation of the elements of th...
Article
Theorem 2.2 in [1] contains an error which has some consequences on its applications in some proofs. In this corrigendum we fix the error and some typos and explain how to fix the proofs.
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The problem of describing the group of units $\mathcal{U}(\mathbb{Z} G)$ of the integral group ring $\mathbb{Z} G$ of a finite group $G$ has attracted a lot of attention and providing presentations for such groups is a fundamental problem. Within the context of orders, a central problem is to describe a presentation of the unit group of an order $\...
Article
We give a new self-contained proof of Poincar\'e's Polyhedron Theorem on presentations of discontinuous groups of isometries of a Riemann manifold of constant curvature. The proof is not based on the theory of covering spaces, but only makes use of basic geometric concepts. In a sense one hence obtains a proof that is of a more constructive nature...
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The title ``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 finite fields and over subfields of finite cyclotomic extensions of the rationals. It also contains functions that produce the primi...
Article
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In this paper we have proposed a model for the distribution of allelic probabilities for generating populations as reliably as possible. Our objective was to develop such a model which would allow simulating allelic probabilities with different observed truncation and de- gree of noise. In addition, we have also introduced here a complete new appro...
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Let R be a commutative ring of characteristic zero and G be an arbitrary group. In the present paper we classify the groups G for which the set of symmetric elements with respect to the classical involution of the group ring RG is Lie metabelian.
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Let $R$ be a commutative ring of characteristic zero and $G$ an arbitrary group. In the present paper we classify the groups $G$ for which the set of symmetric elements with respect to the classical involution of the group ring $RG$ is Lie metabelian.
Article
Let $G$ be a finite group, $\Z G$ the integral group ring of $G$ and $\U(\Z G)$ the group of units of $\Z G$. The Congruence Subgroup Problem for $\U(\Z G)$ is the problem of deciding if every subgroup of finite index of $\U(\Z G)$ contains a congruence subgroup, i.e. the kernel of the natural homomorphism $\U(\Z G) \rightarrow \U(\Z G/m\Z G)$ for...
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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 involu...
Article
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A Z(2)Z(4)Q(8)-code is the binary image, after a Gray map, of a subgroup of Z(2)(k1) x Z(4)(k2) x Q(8)(k3), where Q(8) is the quaternion group on eight elements. Such Z(2)Z(4)Q(8)-codes are translation invariant propelinear codes as are the well known Z(4)-linear or Z(2)Z(4)-linear codes. In this paper, we show that there exist "pure" Z(2)Z(4)Q(8)-...
Article
We give an algorithm to determine a finite set of generators of the unit group of an order in a non-split classical quaternion algebra over an imaginary quadratic extension K of the rationals. We then apply this method to obtain a presentation for the unit group of . As a consequence a presentation is discovered for the orthogonal group . These res...
Article
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 center of the units of the integral group ring $\Z G$. It was recently proved that if $G$ is solvable then there is a Bass unit or a bicyclic unit $v$ and a positive integer $n$ such that the group generated by...
Article
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A Z2Z4Q8-code is a non-empty subgroup of a direct product of copies of Z_2, Z_4 and Q_8 (the binary field, the ring of integers modulo 4 and the quaternion group on eight elements, respectively). Such Z2Z4Q8-codes are translation invariant propelinear codes as the well known Z_4-linear or Z_2Z_4-linear codes. In the current paper, we show that ther...
Article
We compute the rank of the group of central units in the integral group ring $\Z G$ 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 $\Z G$ for a class of groups $G$ properly contained in the finite strongly monomial groups....
Article
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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...
Article
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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 $\Z G$ 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. Th...
Article
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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 rational group algebra QG. This conjecture has been proved for nilpotent groups, metacyclic groups and some other families of groups. We prove the conjecture for cycli...
Article
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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 $\Z G$ generate a subgroup of finite index in its units group $\U(\Z G)$. Our proof provides algorithms to represent some units that contribute to only one simple component of $\Q G$ and generate...
Article
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Marciniak and Sehgal showed that if u is a non-trivial bicyclic unit of an integral group ring then there is a bicyclic unit v such that u and v generate a non-abelian free group. A similar result does not hold for Bass cyclic units of infinite order based on non-central elements as some of them have finite order modulo the center. We prove a theor...
Article
We give a list of finite groups containing all finite groups $G$ such that the group of units $\Z G^*$ of the integral group ring $\Z G$ is subgroup separable. There are only two types of these groups $G$ for which we cannot decide wether $ZG^*$ is subgroup separable, namely the central product $Q_8 Y D_8$ and $Q_8\times C_p{with} p \text{prime and...
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It is shown that ring isomorphisms between cyclic cyclotomic algebras over cyclotomic number fields are essentially determined by the list of local Schur indices at all rational primes. As a consequence, ring isomorphisms between simple components of the rational group algebras of finite metacyclic groups are determined by the center, the dimension...
Conference Paper
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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.
Article
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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 \mathbbFG ® \mathbbFn{\mathbb{F}G \rightarrow \mathbb{F}^n} which maps G to the standard basis of \mathbbFn{\mathbb{F}^n} . Many classical linear codes have been shown to be group codes. In this paper we obtain a criterion...
Article
We present an algorithm to compute the Wedderburn decomposition of semisimple group al- gebras based on a computational approach of the Brauer-Witt theorem. The algorithm was implemented in the GAP package wedderga.
Article
A group code structure of a linear code is a description of the code as one-sided or two-sided ideal of a group algebra of a finite group. In these realizations, the group algebra is identified with the ambient space, and the group elements with the coordinates of the ambient space. It is an obvious consequence of the definition that every pr-ary a...
Article
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 an arbitrary rational prime, r. This completes and unifies previous results of Janusz in [G.J. Janusz, The Schur group of an algebraic number field, Ann. of Math. (2) 103 (1976) 253–281] and Pender...
Article
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In 1992 Drinfeld posed the question of finding the set theoretic solutions of the Yang-Baxter equation. Recently, Gateva-Ivanova and Van den Bergh and Etingof, Schedler and Soloviev have shown a group theoretical interpretation of involutive non-degenerate solutions. Namely, there is a one-to-one correspondence between involutive non-degenerate sol...
Conference Paper
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We present an intrinsecal characterization of when a linear code C is a (left) group code, i.e. the ambient space can be identified with a group algebra in which the standard basis is the group basis such that C is a (left) ideal in this group algebra. As application we obtain a class containing properly the class of metacyclic groups such that eve...
Article
The algebras of Kleinian type are finite-dimensional sernisimple rational algebras A such that the group of units of an order in A is commensurable with a direct product of Kleinian groups. 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...
Article
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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.
Article
We classify the finite groups G such that the group of units of the integral group ring ZG has a subgroup of finite index which is a direct product of free-by-free groups.
Article
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In this note we prove a criteria for the existence of a globalization for a given partial action of a group on an s-unital ring. If the globalization exists, it is unique in a natural sense. This extends the globalization theorem from Dokuchaev and Exel, 2005, obtained in the context of rings with 1.
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In this note we prove a criteria for the existence of a globalization for a given partial action of a group on an s-unital ring. If the globalization exists, it is unique in a natural sense. This extends the globalization theorem from Dokuchaev and Exel, 2005, obtained in the context of rings with 1.
Article
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Let $G$ be a finite group and $\Z G$ its integral group ring. We show that if $\alpha$ is a non-trivial bicyclic unit of $\Z G$, then there are bicyclic units $\beta$ and $\gamma$ of different types, such that $\GEN{\alpha,\beta}$ and $\GEN{\alpha,\gamma}$ are non-abelian free groups. In case that $G$ is non-abelian of order coprime with 6, then we...
Article
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We investigate the group of automorphisms Aut(double-struck QG) of a rational group algebra double-struck QG of a finite metacyclic group G by first describing the simple components of the Wedderburn decomposition of double-struck QG and then investigating when two of these simple components are isomorphic.
Article
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We prove that any projective Schur algebra over a field $K$ is equivalent in $Br(K)$ to a radical abelian algebra. This was conjectured in 1995 by Sonn and the first author of this paper. As a consequence we obtain a characterization of the projective Schur group by means of Galois cohomology. The conjecture was known for algebras over fields of po...
Article
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We prove a conjecture of Zassenhaus that every normalized torsion unit of the integral group ring ZG of a finite group G is rationally conjugate to a group element for some metabelian groups including metacyclic groups G containing a normal cyclic group A such that G/A is cyclic of prime power order. The relative prime case was done in [11].
Article
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We investigate the circumstances under which an inner automorphism of a ring with local units can be built from "local" information. Specifically, we consider three natural "inner" type properties for an automorphism of a ring with local units. We show that every inner automorphism is locally inner but the converse is false, even if the automorphis...
Article
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We obtain a presentation of the group of units of the integral group ring of the group D 16 - =〈a,b∣a 8 =1=b 2 , ba=a 3 b〉.
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We explore a method to obtain presentations of the group of units of an integral group ring of some finite groups by using methods on Kleinian groups. We classify the nilpotent finite groups with central commutator for which the method works and apply the method for two concrete groups of order 16.
Article
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For a unital ring R, $RCFM_{\alpha}(R)$ denotes the ring of row and column finite matrices over R indexed by α. We give necessary and sufficient structural conditions on $RCFM_{\alpha}(R)$ which are equivalent to R being, respectively, Quasi-Frobenius, left artinian, and left noetherian.
Article
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We explore a method to obtain presentations of the group of units of an integral group ring of some finite groups by using methods on Kleinian groups. We classify the nilpotent finite groups with central commutator for which the method works and apply the method for two concrete groups of order 16. D.G.I. of Spain and Fundación Séneca of Murcia. AM...
Article
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We give a method to obtain the primitive central idempotent of the rational group algebra QG over a finite group G associated to a monomial irreducible character which does not involve computations with the character field nor its Galois group. We also show that for abelian-by-supersolvable groups this method takes a particularly easy form that can...
Article
We present an algorithm to compute the primitive central idempotents and the Wedderburn decomposition of a rational group algebra. This algorithm has been implemented for System GAP, version 4.
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We prove that the group generated by the bicyclic units of ZS<sub>n</sub> has torsion for n≥ 4 This answers a question of Sehgal (1993).
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We prove that the group generated by the bicyclic units of Z S n \mathbb {Z} S_n has torsion for n ≥ 4 n\ge 4 . This answers a question of Sehgal (1993).
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In this note the authors correct and extend results presented in their article The Isomorphism problem for incidence rings, Pacific J. Math., 187(2) (1999), 201-214. Specifically, it is shown that for a large class of rings (including those with finite right Goldie dimension, semilocal, and many commutative rings), if P and P' are finite preordered...
Article
is semisimple (meaning in this paper semisimple artinian) has been studied byseveral authors. The most classical result is Maschke’s Theorem for group rings.For crossed products over fields there is a satisfactory answer given by Aljadeffand Robinson [3]. Another partial answer for skew group rings was given byAlfaro et al. [1]. A reduction of the p...
Article
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In [5] Ritter and Sehgal introduced the following units, called the bicylic units, in the unit group U(ZG) of the integral group ring ZG of a finite group G: ¯a;g = 1 + (1 ¡ g)abg; °a;g = 1 + bga(1 ¡ g); where a; g 2 G and bg is the sum of all the elements in the cyclic group hgi. It has been shown that these units generate a large part of the unit...
Article
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We construct explicitly a subgroup of finite minimal index and minimal rank in which is a direct products of free groups for each finite group G for which this is possible The first author has been partially supported by the DGI of Spain and Fundación Seneca of Murcia
Article
In this note the authors correct and extend results presented in their article "The Isomor- phism problem for incidence rings", Pacific J. Math 187(2), 1999, 201-214. Specifically we show that for a large class of rings (including those with finite right Goldie dimension, semilocal, and many commutative rings), if P and P0 are finite preordered set...
Article
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We show that isomorphism of intermediate rings between row and column finite matrix rings and row finite matrix rings implies Morita equivalence of the coefficient rings and equality of the cardinality of the set of indices. Among the applications we extend the Isomorphism Theorem for Dual Pairs over Division Rings to Ornstein dual pairs over any c...
Article
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We classify the finite groups G such that the unit group of the in-tegral group ring ZG contains a subgroup of finite index which is the direct product of free products of abelian groups. The latter prob-lem is also investigated for any order in an arbitrary semisimple finite dimensional Q-algebra.
Article
We study two subgroups of the Picard group Pic(RGr) of a category of graded modules: the subgroup gr−Pic(R#PG) of classes of graded equivalences and the subgroup Pic(R#PG)G of classes of equivalences which commute with the suspensions in Pic(RGr). We prove that in general these groups are not the same and show that they are related by an exact sequ...
Article
Let P and P′ be finite preordered sets, and let R be a ring for which the number of nonzero summands in a direct decomposition of the regular module RR is bounded. We show that if the incidence rings I(P, R) and I(P′, R) are isomorphic as rings, then P and P′ are isomorphic as preordered sets. We give a stronger version of this result in case P and...
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For a ring A with local units we investigate unital overrings T of A, and compare the automorphism groups Aut(A) and Aut(T).
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This paper investigates the ring-theoretic similarities and the cate- gorical dissimilarities between the ring RFM(R) of row nite matrices and the ring RCFM(R) of row and column nite matrices. For example, we prove that two rings R and S are Morita equivalent if and only if the rings RCFM(R) and RCFM(S) are isomorphic. This resembles the result of...
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We classify all the finite groupsG, such that the group of units ofZGcontains a subgroup of finite index which is isomorphic to a direct product of nonabelian free groups. This completes the work of8, where the nilpotent groups with this property are given.
Article
We classify the nilpotent finite groupsGwhich are such that the unit group U(ZG) of the integral group ringZGhas a subgroup of finite index which is the direct product of noncyclic free groups. It is also shown that nilpotent finite groups having this property can be characterised by means of the Wedderburn decomposition of the rational group algeb...
Article
For R a G-graded ring, we study Pic(R-gr), the group of isomorphism classes of autoequivalences of the category of graded left R-modules. For G infinite, this requires generalizing the classical sequences involving Pic(A), A a fc-algebra, to A a ring with local units. Then for G either finite or infinite, we characterize the inner automorphisms in...
Article
We study the relation between the injective dimension of a ring as a module over itself and that of the fixed ring under some group of automorphisms of the ring. In special cases we show how to derive an injective resolution of the fixed ring from an injective resolution of the original ring.
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Let S be a semigroup and let be an S -graded ring. R s = 0 for all but finitely many elements s ∈ S 1 , then R is said to have finite support . In this paper we concern ourselves with the question of whether a graded ring R with finite support inherits a given ring theoretic property from the homogeneous subrings R e corresponding to idempotent sem...
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We present a short history of the following problem: Classify the finite groups G, so that the group of units of the integral group ring ℤG contains a subgroup of finite index isomorphic to a direct product of non-Abelian free groups.
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Let G be a group acting on a ring R . We study the problem of determining necessary and sufficient conditions in order that the skew group ring RG be von Neumann regular. Complete characterizations are given in some particular situations, including the case where all idempotents of R are central. For a regular ring R admitting a G -invariant pseudo...
Article
We present a collection of finitely generated projective generators for the category of graded modules over a unital semigroup-graded ring. Consequences of the existence of such a collection, as well as other module-theoretic re-sults, will arise from more general constructions involving TTF classes inside Grothendieck categories. For instance, we...

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