icm 2006 posters abstracts section 02

ICM 2006

Posters

Abstracts

Section 02Algebra

ICM 2006 – Posters. Abstracts. Section 02

On generalized left derivations in rings

Shakir Ali

Department of Mathematics, Aligarh Muslim University, Aligarh-202002, [email protected]

2000 Mathematics Subject Classification. 16W25, 16N60, 16U80

Throughout the discussion, unless otherwise mentioned, R denotes an asso-ciative ring(may be without unity)with center Z(R). An additive mappingF : R −→ R is called a generalized derivation on R if there exists a deriva-tion d : R −→ R such that F (xy) = F (x)y+xd(y), holds for all x, y ∈ R. Anadditive mapping G : R −→ R is called a generalized left derivation on R ifthere exists a left derivation δ : R −→ R such that G(xy) = xG(y) + yδ(x),holds for all x, y ∈ R.

In this article, we introduce the notion of generalized left derivationsand obtained some recent results on generalized derivations to generalizedleft derivation. The main result state as follows:Theorem. Let R be a prime ring and I a non-zero ideal of R. Then thefollowing conditions are equivalent:(i) If R admits a generalized left derivation G associated with a non-zeroleft derivation δ such that G(xy) − xy ∈ Z(R) or G(xy) + xy ∈ Z(R) forall x, y ∈ I(ii) If R admits a generalized left derivation G associated with a non-zeroleft derivation δ such that G(xy) − yx ∈ Z(R) or G(xy) + yx ∈ Z(R) forall x, y ∈ I(iii) R is commutative.

In addition, we also established some related results and discuss someexamples which demonstrates that R to be prime is essential in the hypoth-esis of our result.

ICM 2006 – Madrid, 22-30 August 2006 1


On universal central extensions of precrossed and crossedmodules

Daniel Arias*, Jose Manuel Casas, Manuel Ladra

Departamento de Matematicas, Universidad de Leon, Campus de Vegazana,Leon, E-24071, [email protected]

2000 Mathematics Subject Classification. 20J05, 18G50

The notion of universal extension was introduced by Kervaire in [4]. Heshowed that for every perfect group G there is a central extension 0 →H2(G) → U → G → 1 such that there is a unique map from it to eachcentral extension of G.

In general, universal central extensions in the categories PCM of pre-crossed modules and CM of crossed modules don’t coincide, though bothof them contain as second component the universal extension of Kervaire(see [1]).

We deal with the interpretation of their first components, by comparingthem with universal extensions in the categories of epimorphisms of groups[5], groups with operators [3] and modules [2].

We also establish in which conditions universal central extensions inPCM and CM coincide.

Finally we classify the perfect precrossed modules with the same univer-sal extension. As an application, we show how to lift an automorphism ofa perfect precrossed module (M,P, µ) to an automorphism of its universalextension U(M,P, µ), which allows to consider the automorphisms groupAut(M,P, µ) as a subgroup of Aut(U(M,P, µ)).

Each result is applied to the following example from algebraic K-theory:for a two-sided ideal I of a ringR there is a perfect crossed module (E(I), E(R), i)of elementary matrices, with universal central extension in PCM (K2(I),K2(R), γ) �(St(I), St(R), γ) � (E(I), E(R), i), while its universal central extensionin CM is (K2(R, I),K2(R), γ) � (St(R, I), St(R), γ) � (E(I), E(R), i),where St(I) and K2(I) are the Stein relativizations of St(R) and K2(R),and St(R, I) and K2(R, I) denote the relative groups of Loday and Keune(see [1] or [5]).

References

[1] Arias, D., Ladra, M., R.-Grandjean, A., Universal central extensions of pre-crossed modules and Milnor’s relative K2, J. Pure Appl. Algebra 184 (2003),139–154.

2 ICM 2006 – Madrid, 22-30 August 2006


[2] Blanc, D., Peschke, G., The plus construction, Postnikov towers and universalcentral module extensions, Israel J. Math. 132 (2002), 109–123.

[3] Cegarra, A. M., Inassaridze, H., Homology of groups with operators, Intern.Math. J. 5 (1) (2004), 29–48.

[4] Kervaire, M. A., Multiplicateurs de Schur et K-theorie. In Essays on Topologyand Related Topics (Memoires dedies a G. de Rham). Springer, Berlin 1970,212–225.

[5] Loday, J.-L., Cohomologie et groupes de Steinberg relatifs, J. Algebra 54 (1)(1978), 178–202.



Solution to Serre’s injectivity problems in non-abelian galoiscohomology of algebraic groups

Pedro Benjamin Barquero-Salavert

2000 Mathematics Subject Classification. 20G15, 11E72, 11Exx, 16K20,17Axx

For an algebraic group G over a field F, we will write H1(F,G) for the firstcohomology set H1(Γ, G(Fsep)), where Fsep is the separable closure of F,G(Fsep) is the abstract group of Fsep−points and Γ = Gal(Fsep/F ) is theabsolute galois group of F. The pointed set H1(F,G) is functorial in both Gand F. If L/F is a field extension, there is a natural induced map of pointedsets H1(F,G) −→ H1(L,G). In Serre (1997, p. 192), the author asks forwhat algebraic groups and field extensions this induced map is injective.Several cases were already known, for instance:

1) G = O(V, q), the orthogonal group of a quadratic space (V, q), andL/F an odd-degree extension. (T. Springer, 1950’s)

2) GU(V, h), the unitary group of a hermitian module (V, h) and L/Fan odd-degree extension (Bayer-Lenstra, 1990)

We present the proof that the map of pointed setsH1(F,G) −→ H1(L,G)is injective in these following other cases, the first three of which have al-ready been published [1]:

1) G = PGO(A, σ), the projective general orthogonal group of a centralsimple algebra with orthogonal involution and L/F an odd-degree exten-sion, where char(F ) 6= 2.

2) G = PGSp(A, σ), the projective general symplectic group of a centralsimple algebra with symplectic involution and L/F an odd-degree extension,where the basefield F is arbitrary.

3) G = PGU(B, τ), the projective general unitary group of a centralsimple algebra with unitary involution and L/F an odd-degree extension,where the basefield F is arbitrary.

4) G is a simple group of exceptional type G2, and L/F an odd-degreeextension of degree coprime to 3, where the basefield F is arbitrary.

5) G is a simple group of exceptional type F4, and L/F an odd-degreeextension of degree coprime to 3, where the basefield F is arbitrary.

6) G is a simple simply connected group of exceptional type E7, andL/F an odd-degree extension of degree coprime to 3, where char(F ) 6= 2, 3.

I wish to enter the poster competition.



References

[1] P. B. Barquero-Salavert, Similitudes of Algebras with Involution under odd-degree extensions. In Communications in Algebra, vol 34-2, 2006

[2] N. Bourbaki, Algebra, livre II, chapitre IX, Hermann, Paris, 1959.

[3] N. Jacobson, Structure and Representations of Jordan Algebras, ColloquiumPublications, vol. 39, AMS, 1968.

[4] M. Knus et al., The Book of Involutions, Colloquium Publications, vol. 44,AMS, 1998.

[5] J. P. Serre, Galois cohomology. LNM. Springer-Verlag. Berlin. 1997.



Some generalisations of the notion of local formation

A. Ballester-Bolinches, Clara Calvo∗, and R. Esteban-Romero

Departament d’Algebra, Universitat de Valencia, Dr. Moliner, 50, 46100Burjassot, Valencia, Spain; Departament de Matematica Aplicada-IMPA,Universitat Politecnica de Valencia, Camı de Vera, s/n, 46022 Valencia, [email protected]; [email protected]; [email protected]

2000 Mathematics Subject Classification. 20D10, 20F17

A formation is a class of groups which is closed under epimorphic imagesand subdirect products.

The notion of X-local formation, where X is a class of simple groups witha completeness property, was introduced by P. Forster in [1]. It generalisesat the same time the definitions of local formation and Baer-local formation.It is possible to define a Frattini-type subgroup associated with a class X

which allows us to characterise the X-local formations in terms of an X-saturation.

Local formations can be also seen as a particular case of ω-local forma-tions, where ω is a set of primes. However, X-local and ω-local formationsbehave differently.

We will show in this poster some results relating both concepts. Forinstance, given an X-local formation F, there exists an ω-local formationG, where ω is the characteristic of X, such that G is the largest ω-localformation contained in F. As a corollary, we obtain that ω-local and X(ω)-local formations coincide in the ω-separable universe, where X(ω) is theclass of all simple groups whose order is divisible by primes in ω.

References

[1] Forster, P., Projektive Klassen endlicher Gruppen IIa. Gesattigte Formationen:ein allgemeiner Satz von Gaschutz-Lubeseder-Baer-Typ. Publ. Sec. Mat. Univ.Autonoma Barcelona, 29 (1985), 39–76.

[2] Ballester-Bolinches, A., Calvo, C., Esteban-Romero, R., A question of theKourovka Notebook on formation products, Bull. Austral. Math. Soc., 68(2003), 461–470.

[3] Ballester-Bolinches, A., Calvo, C., Esteban-Romero, R., On X-saturated for-mations of finite groups, Commun. Algebra, 33 (2005), 1053–1064.

[4] Ballester-Bolinches, A., Calvo, C., Shemetkov, L. A., On partially saturatedformations of finite groups, preprint.



Obsruction theory of Lie–Rinehart algebras

J. M. CasasDpto. Matematica Aplicada I, E.U.I.T. Forestal, Universidad de Vigo, CampusUniversitario A Xunqueira, 36005 Pontevedra, [email protected]

2000 Mathematics Subject Classification. 17B55, 17B56, 17B60

Lie–Rinehart algebras are the algebraic counterpart of Lie algebroids [3]and they play an important role in many branches of mathematics [2]. ALie–Rinehart algebra over a K-commutative algebra A consists with a LieK-algebra L together with an A-module structure on L and a map, calledanchor map, α : L → Der(A) which is simultaneously Lie algebra andA-module homomorphism such that the following relation holds

[X, aY ] = a[X,Y ] +X(a)Y

for any X,Y ∈ L, a ∈A (X(a) means α(X)(a)).A triple cohomology theory for Lie–Rinehart algebras was developed

in [1]. This cohomology is isomorphic to the Rinehart cohomology [2, 4]provided than L is projective as A-module. Moreover the low dimensionalcohomology groups were interpreted by means of abelian and crossed ex-tensions.

In the present work an eight-term exact sequence in cohomology asso-ciated to an epimorphism of Lie–Rinehart algebras is obtained. Technicalproblems as the construction of the actor object in the category of Lie–Rinehart algebras are studied and abelian induced extensions and (L, A)-crossed extensions are introduced. We apply these results to study obstruc-tion theory of Lie–Rinehart algebras extensions, achieving the generaliza-tion of the classical results:

a) An abstract kernel is realized by an extension if and only if its ob-struction class is zero.

b) If the obstruction class of an abstract kernel ψ : P → OutDO(A,L, N)is zero, then there exists a one-to-one correspondence between the set of ex-tensions of P by N associated to the abstract kernel ψ and H2(A,L, Z(N)).

In particular case A = K we recover the classical results in Lie algebrastheory.

References

[1] Casas, J. M., Ladra, M. , Pirashvili, T., Triple cohomology of Lie-Rinehartalgebras and the canonical class of associative algebras, J. of Algebra 291(1)(2005), 144–163.



[2] Huebschmann, J. , Poisson cohomology and quantization, J. Reine Angew.Math. 408 (1990), 57–113.

[3] Mackenzie, K., Lie groupoids and Lie algebroids in Differential Geometry. Lon-don Math. Soc. Lecture Note Ser. 124, Cambridge University Press, 1987.

[4] Rinehart, G. S., Differential forms on general commutative algebras, Trans.Amer. Math. Soc., 108 (1963), 195–222.



Derivations of a restricted Weyl type algebra I

Seul Hee Choi

Dept. of Mathematics, Jeonju University, Chon-ju 560-759, [email protected]

2000 Mathematics Subject Classification. 17B40, 17B56

One of the evaluation algebra which is defined by the author and coworkersis the Weyl type non-associative algebra WNn,m,sr. The non-associativealgebra WNn,m,sr contains the matrix ring Mm+s(F ), the polynomial ringF [x1, · · · , xs], and the Witt algebra W (s). All the derivations of the Weylalgebra, the polynomial ring, and the Witt type algebra are well known.We find all the derivations of WN0,0,s1 and WN0,s,01, and also list someopen problems on some F -algebra, Lie algebra, semi-Lie algebra, and non-associative algebra.



Extending modules versus purely extending modules

Septimiu Crivei

Faculty of Mathematics and Computer Science, ”Babes-Bolyai” University, Str.Mihail Kogalniceanu 1, 400084 Cluj-Napoca, [email protected]

2000 Mathematics Subject Classification. 16D

Extending modules (or CS modules) are defined as modules with the prop-erty that every submodule is essential in a direct summand [2]. They havebeen intensively studied throughout the last two decades. In 1995 L. Fuchs[3] considered their generalization by replacing ”direct summand” with”pure submodule” in the definition. Thus one obtains the so-called purelyextending modules, whose study was continued by J. Clark [1]. So far theresults on purely extending modules are few, mainly because of the ”gap”between direct summands and pure submodules. We review and show newproperties of purely extending modules as well as introduce some interme-diate classes of modules between extending and purely extending modules,whose study is easier to be done. It has been observed that projective mod-ules and, more generally, the class Add(M) of all summands of direct sumsof copies of a module M , play an important part in the study of extendingmodules. In our study we make use of some classes of modules generalizingthe class Add(M) and establish characterizations of modules generalizingmodules with the property that any direct sum of copies is extending.

References

[1] Clark, J., On purely extending modules, In Abelian groups and modules. Pro-ceedings of the international conference in Dublin, Ireland, August 10-14, 1998(ed. by Eklof, Paul C. et al.), Basel, Birkhauser, Trends in Mathematics, 1999,353–358.

[2] Dung, N.V., Huynh, D.V., Smith, P.F., Wisbauer, R., Extending modules,Pitman Research Notes in Mathematics Series, 313, Longman Scientific &Technical, 1994.

[3] Fuchs, L., Notes on generalized continuous modules, preprint, 1995.

[4] Gomez Pardo, J.L., Guil Asensio, P.A., Indecomposable decompositions ofmodules whose direct sums are CS, J. Algebra 262 (2003), 194–200.



On some classes of supersoluble groups

A. Ballester-Bolinches, J. C. Beidleman, and R. Esteban-Romero*

Departament d’Algebra, Universitat de Valencia, Dr. Moliner, 50, E-46100Burjassot, Valencia, Spain; Department of Mathematics, University of Kentucky,Lexington, Kentucky 40506-0027, USA; Departament de MatematicaAplicada-IMPA, Universitat Politecnica de Valencia, Camı de Vera, s/n, 46023Valencia, [email protected]; [email protected]; [email protected]

2000 Mathematics Subject Classification. 20D10, 20D20, 20F16

This work is dedicated to professor Hermann Heineken on the occasion ofhis seventieth birthday.

Finite groups G for which for every subgroup H and for all primes qdividing the index |G : H| there exists a subgroup K of G such that H iscontained in K and |K : H| = q are called Y-groups. These groups can beseen as satisfying a converse of the well-known Lagrange theorem. Y-groupsare supersoluble. A study of these groups appears in [2].

On the other hand, groups in which subnormal subgroups permute withall Sylow subgroups are called PST-groups. This class of groups generalisesthe class of T-groups or groups in which normality is a transitive relation.PST-groups have been studied by many authors (see, for instance, [1, 3, 4]).

In this poster a local version of the Y-property leading to a local charac-terisation of Y-groups, from which the classical characterisation emerges, isintroduced. This local characterisation runs parallel to the local characteri-sations of PST-groups analysed in [3]. The relationship between PST-groupsand Y-groups is also analysed: A group G is a soluble PST-group if and onlyif G is a Y-group and the nilpotent residual of G is abelian. Similar resultshold for the local characterisations.

References

[1] Agrawal, R. K., Finite groups whose subnormal subgroups permute with allSylow subgroups, Proc. Amer. Math. Soc., 47 (1975), 77–83.

[2] Ballester-Bolinches, A., Beidleman, J. C., Esteban-Romero, R., On some classesof supersoluble groups, to appear.

[3] Ballester-Bolinches, A., Esteban-Romero, R., Sylow permutable subnormalsubgroups of finite groups II, Bull. Austral. Math. Soc., 64 (2001), 479–486

[4] Beidleman, J. C., Heineken, H, Finite soluble groups whose subnormal sub-groups permute with certain classes of subgroups. J. Group Theory, 6 (2003),139–158.



[5] Weinstein, M., editor, Between nilpotent and solvable, Polygonal PublishingHouse, Passaic, NJ, USA, 1982.



Arithmetical conditions on conjugacy class sizes in finite groups

Marıa Jose Felipe Roman*, Antonio Beltran Felip

Departamento de Matematica Aplicada, Universidad Politecnica de Valencia,Spain; Departamento de Matematicas, Universidad Jaume I de Castellon, [email protected]; [email protected]

2000 Mathematics Subject Classification. 20E45

A classic problem in Finite Group Theory is to study the structure of agroup G from the set of its conjugacy class sizes, cs(G). Some known the-orems determine the non-simplicity, solvability or nilpotency of G, whensome arithmetical conditions are imposed on cs(G), but often the proofsmake use of the Classification of the Finite Simple Groups, or deep tech-niques of Ordinary an Modular Representation Theory.

Among these results we can mention the works of N. Ito, who estab-lished the nilpotency of groups with |cs(G)| = 2, the solvability of groupswith |cs(G)| = 3 and also determined all simple groups with |cs(G)| = 4. Onthe other hand, A. Camina showed ([3]) that when cs(G) = {1, pa, qb, paqb}where p and q are two distinct primes, then G is nilpotent. In this con-text, we expose several properties obtained in the case |cs(G)| = 4. Moreprecisely, we have proved by using elementary methods that groups withcs(G) = {1,m, n,mn}, with m and n two coprime numbers are nilpotent([1], [2]). Likewise, we get several properties on the structure of solvablegroups satisfying cs(G) = {1,m, n,mk}, with (m,n) = 1 and k a properdivisor of n. Also, we present some results extended for p-regular classes,that is, for conjugacy classes of elements having p′-order. We obtain thatp-solvable groups with four conjugacy p-regular class sizes and satisfyingsome additional arithmetical conditions posses nilpotent p-complements.

References

[1] Beltran, A., Felipe, M.J. Variations on a theorem by Alan Camina on conjugacyclass sizes, J. Algebra. 296 (2006), 253–266.

[2] Beltran, A., Felipe, M.J. Some class size conditions implying solvability of finitegroups. Accepted in J. Group Theory.

[3] Camina, A.R. Arithmetical conditions on the conjugacy class numbers of afinite group, J. London Math. Soc. 2 (5) (1972) 127–132.



On p-groups of maximal class

M. A. Garcıa-Sanchez

Dpto. Matematicas, Fc. Ciencia y Tecnologıa, UPV-EHU, Apto. 644, 48080Bilbao, Spain20D15

2000 Mathematics Subject Classification. 20D15

The first major study of p-groups of maximal class was made by Blackburnin 1958 (cf.[1]). He showed that an important invariant parameter of thesegroups is its ‘degree of commutativity’, which is a measure of the commuta-tivity among the members of the lower central series of G. It is interesting toobtain a lower bound for the degree of commutativity of a p-group of max-imal class, because the greater the bound is, the lower number of variablesappears in the commutator structure of G.

In order to obtain good lower bounds, we know that certain invariantparameters of G are necessary. If we only consider as parameters p, m andc(G), where |G| = pm and c(G) is the degree of commutativity of G, weknow that we lose information (cf. [3]). However, if we consider two moreinvariant parameters, concretely c0(G), the residual class of c(G) modulep−1 and l, defined in [2], the lower bound for the degree of commutativity ismore accurate. In this poster, we summarize the best obtained lower boundfor the degree of commutativity of p-groups of maximal class, in terms ofp, m, c(G), c0(G) and l. The expression of the bound changes, accordingto the conditions on l and c0(G) that are satisfied in G. These expressionshave been showed in [3], [4] and [5]. As a corollary, we obtain an upperbound for the number of non-isomorphic p-groups of maximal class. Thisbound improves the best known upper bound for these groups. Finally, wegive information about the nilpotence class of G1.

References

[1] N. Blackburn, On a special class of p-groups, Acta Math., 100, 1958, 45–92

[2] A. Vera-Lopez, J. M. Arregi, and F. J. Vera-Lopez, Some bounds on thedegree of commutativity of a p-group of maximal class, II. Comm. in Algebra,23:2765–2795, 1995.

[3] A. Vera-Lopez, J. M. Arregi, M.A. Garcıa-Sanchez, R. Esteban-Romero andF. J. Vera-Lopez, The exact bound for the degree of commutativity of a p-groupof maximal class, I. J. Algebra, 256 (2002) 375–401.



[4] A. Vera-Lopez, J. M. Arregi, M.A. Garcıa-Sanchez, R. Esteban-Romero andF. J. Vera-Lopez, The exact bound for the degree of commutativity of a p-groupof maximal class, II. J. Algebra, 273 (2004) 806–853.

[5] M.A. Garcıa-Sanchez, A. Vera-Lopez, J. M. Arregi and L. Ormaetxea, Onthe degree of commutativity of a p-group of maximal class. submitted forpublication.



Noetherian semigroup algebras and maximal orders

Isabel Goffa* and Eric Jespers

Department of Mathematics, Free University of Brussels (VUB), Pleinlaan 21050 Brussels, [email protected]

2000 Mathematics Subject Classification. 20M25

The only groups for which it is known that the group algebra is Noetherianare the polycyclic-by-finite groups. Brown characterised when these groupalgebras are prime maximal orders ([1]). In the search for other Noetherianrings one started to look at semigroup algebras. There we know that thesemigroup algebra of an abelian monoid is Noetherian if and only if themonoid is finitely generated [2]. However, in general this is unknown. In [5],the authors characterised when a submonoid of a torsion-free finitely gen-erated abelian-by-finite group has a semigroup algebra that is a Noetherianmaximal order. It turned out that the behavior of the minimal primes ofthe semigroup are very important.

We extend these results to the non torsion-free and polycyclic-by-finitecase, and we prove when special semigroups are Noetherian maximal orders.For example, we generalise the notion of a monoid of I-type (which has asemigroup algebra that is a Noetherian maximal order and a domain, [4]).We call these monoids the ‘Monoids of IG-type’ and we prove that this largeclass often has a nice algebraic structure too ([3]).

References

[1] K. A. Brown, Height one primes of polycyclic group rings, London, Math. Soc.(1985), 426–438.

[2] Gilmer R., Commutative semigroup rings, Univ.Chicago Press, Chicago, 1984.

[3] I. Goffa and E. Jespers, Monoids of IG-type and Maximal Orders, submittedto J.Algebra

[4] E. Jespers and J. Okninski, Monoids and groups of I-type, Algebras Repres.Theory 8 (2005), 709–729.

[5] E. Jespers, J. Okninski, Semigroup algebras and Noetherian maximal orders,J. Algebra 238 (2001), 590–662.



Solving systems of linear equations uniformly by generalizingCramer’s Rule

G. M. Diaz–Toca∗, L. Gonzalez–Vega+, H. Lombardi++

Dpto. de Matematica Aplicada, Universidad de Murcia, 30071 Murcia, Spain;+Dpto. de Matematicas, Estadıstica y Computacion, Universidad de Cantabria,39005 Santander, Spain; ++Laboratoire de Mathematiques, Universite deFranche–Comte, 25030 Besancon, [email protected]

2000 Mathematics Subject Classification. 15A09,15A06,68W30

Given a system of linear equations over an arbitrary field K with coefficientmatrix A ∈ Mn,m(K), A · x = b, the existence of solution depends only onthe ranks of the matrices A and A|b. These ranks can be easily obtained byusing Gaussian elimination, but, if the system (the matrix A or the vectorb) depends on parameters then the use of Gaussian elimination makes thisapproach very expensive since it produces a combinatorial explosion whenconsidering the parametric vanishing of the different pivots. Moreover thismethod is not uniform (e.g. the resulting conditions will strongly depend onthe characteristic of K) and not easy to parallelize ([2]). The same behaviouris found when the considered system of linear equations is solved accordingto the different possible values of the parameters.

If A ∈Mn,m(K), by generalizing Mulmuley’s Algorithm ([3]), it is shown([1]) how the coefficients of the characteristic polynomial of the n–squarematrix

A ·A◦ = A · diagonal(1, z, . . . , zm−1) ·At · diagonal(1, z, . . . , zn−1)

(where z is a new variable) make possible in a very uniform and compactway, first, to characterize the rank of A, second and once the rank of Ais fixed, to determine the equations of Im(A) and, given b ∈ Im(A), thesolution of A · x = b generalizing the classical Cramer’s Rule to the under-determined and overdetermined cases.

This provides a generalization of the classical Gram’s coefficients in thereal or complex case ([2]) providing, to the cost of adding a new variable,the surprising fact that for fields of characteristic p and for C the rank ofany matrix can be also controlled by a small number of sums of squaresof minors of the considered matrix which avoids the usual combinatorialexplosion.



References

[1] Diaz–Toca, G. M., Gonzalez–Vega, L., Lombardi, H., Generalizing Cramer’sRule: solving uniformly linear systems of equations, SIAM Journal on MatrixAnalysis and Applications 27 (2005), 621–637.

[2] Abdeljaoued, J., Lombardi, H., Methodes Matricielles: Introduction a la Com-plexite Algebrique, Mathematics & Applications 42. Springer–Verlag, Berlin,2004.

[3] Mulmuley, K., A fast parallel algorithm to compute the rank of a matrix overan arbitrary field, Combinatorica 7 (1987), 101–104.



Rings whose non-zero finitely generated modules are retractable

A. Haghany*, M. R. Vedadi

Department of Mathematics, Isfahan University of Technology, Isfahan, [email protected]; [email protected]

2000 Mathematics Subject Classification. 16S50

We give several equivalent formulations of a finite retractable ring whichis defined to be a ring R, all of whose non-zero finitely generated (right)modules M are retractable, in the sense that HomR(M,N) 6= 0 for any non-zero submoduleN ofM . One such formulation involving matrix rings overRstates that if I is any right ideal in S = Matn×n(R) and x ∈ S\I, then thereexists s ∈ S such that xs 6∈ I and xsI ⊆ I. Initially, retractable modulesappeared in [3] and then in connection with the study of endomorphismrings, being Baer, CS, quasi Baer, etc. in [1],[2],[4]. More recently, P. F.Smith [5] characterized retractable modules over right FBN rings. In thispaper, we use our characterizations of finite retractable rings to show thatthe class C of these rings contains any ring that is Morita equivalent to acommutative ring, and that if R is a right order in T , then R ∈ C impliesthat T ∈ C. Finally, a finitely annihilated module M over a finite retractablering is shown to be a weak generator in σ[MR].

References

[1] Beachy, J. A., M-Injective modules and prime M-ideals, Communications inAlg. 30, no. 10, (2002), 4649-4976.

[2] Haghany, A., Vedadi, M. R., Endoprime modules, Acta Math.Hungar., 106(1-2), (2005), 89-99.

[3] Khuri, S. M., Endomorphism rings and lattice isomorphisms, Journal of Alg.,59, no. 2, (1979), 401-408.

[4] Rizvi, S. T., Roman, C. S., Baer and Quasi-Baer Modules, Communicationsin Alg., 32, no. 1, (2004), 103-123.

[5] Smith, P. F., Modules with many homomorphisms, Journal of Pure and Ap-plied Alg., 197 (1-3), (2005), 305-321.



Homomorphic order and constructive order on operations

Kiyomitsu Horiuchi

Faculty of Science and Engineering, Konan University, Okamoto, Higashinada,Kobe 658-8501, [email protected]

2000 Mathematics Subject Classification. 08A40

We would like to introduce two orders on sets of the operations of thealgebra system. Homomorphism is a fundamental concept in the algebrasystem with the operation. In case of the algebra system with two or moreoperations, the concept of individual homomorphism can be considered re-spectively for each operation. For instance, there are two operations onevery lattice. Hence, there are two homomorphism on every lattice. Thesetwo homomorphism is independent for a general lattice. In case of the to-tally ordered set, two homomorphism are corresponding. In another algebrasystem with two operations or more, there are some cases where the secondoperation is sure to become homomorphism if the first operation is homo-morphism. Moreover, another operation might become homomorphism fromthe homomorphism of two operations or more. We would like to introducethe order on sets of the operations of the algebra system by using this. Wecall this “Homomorphic order.” On the other hand, the operation may beconstructed of other operations. By using this, we can think another orderon sets of the operations of the algebra system. We call this “Constructiveorder.” We study the relations of these orders.

References

[1] S. Burris, H.P.Sankappanavar. A Course in Universal Algebra GTM 78Springer-Verlag, 1981.

[2] G. Birkhoff. Lattice Theory (third ed.) Amer. Math. Soc. Colloq. Publ., 1967.

[3] G. Gratzer. General Lattice Theory (Second ed.) Birkhauser, 1998.

[4] K. Horiuchi. Trice and Two delegates operation. Scientiae Mathematicae, 2-3(1999)373–384.

[5] K. Horiuchi, A. Tepavcevic. On distributive trices. Discussiones MathematicaeGeneral Algebra and Applications, 21(2001)21–29.

[6] K. Horiuchi. Some weak laws on bisemilattice and triple-semilattice. ScientiaeMathematicae, 59, No.1 (2004): 41–61.



Ternary derivations of finite-dimensional real division algebras

Clara Jimenez–Gestal*, Jose Marıa Perez–Izquierdo

Dpto. Matematicas y Computacion, Universidad de La Rioja, Edificio Vives, Luisde Ulloa S/N, 26003 Logrono, [email protected]

2000 Mathematics Subject Classification. 17A35

We present a group acting on the family of finite–dimensional real divisionalgebras and an invariant, the Lie algebra of ternary derivations, for thisaction. An exploration of this family is conducted in terms of this newinvariant obtaining simple descriptions of the division algebras involved.In the course of the investigation, the algebra sl(4, F ) of 4 × 4 tracelessmatrices with the symmetric product xy+yx− 1

2 t(xy)I shows an exceptionalbehavior.

References

[1] G. M. Benkart, J. M. Osborn: The derivation algebra of a real division algebra,Amer. J. Math. 103 (1981), no. 6, 1135–1150.

[2] G. M. Benkart, J. M. Osborn: An investigation of real division algebras usingderivations, Pacific J. Math. 96 (1981), no. 2, 265–300.

[3] D. Z. Dokovic, K. Zhao: Real division algebras with large automorphism group,J. Algebra 282 (2004), no. 2, 758–796.



On finiteness of multiplication modules

Hashem Koohy

Department of Mathematics, The University of Warwick, CV4 7AL, [email protected]

2000 Mathematics Subject Classification. 13E05,13EXX

Our main aim in this note, is a further generalization of a result due toD. D. Anderson, i.e., it is shown that if R be a commutative ring, andM a multiplication R−module, such that every prime ideal minimal overAnn(M) is f.g., then M contains only a finite number of minimal primesubmodules. This immediately yields if P is a projective ideal ofR, such thatevery prime ideal minimal over Ann(P ) is f.g., then P is finitely generated.Furthermore, it is established that if M is a multiplication R−module inwhich every minimal prime submodule is f.g., then R contains only a finitenumber of prime ideals minimal over Ann(M). Finally, an approach is givenfor commutative ue−rings, by classifying modules in which every properessential submodule is prime, i.e., it is shown that every proper essentialsubmodule of M is prime, iff, M/N is very semisimple, for every properessential submodule N of M , iff, every proper essential submodule of M isvirtually maximal, iff, M/soc(M) is very semisimple.

References

[1] D. D. Anderson (1994). A Note on Minimal Prime Ideals, Proc.Amer.Math.Soc., volume 122, number 1, 1994.

[2] . Behboodi, H. Koohy (2002). On Minimal Prime submodules, Far EastJ.Math.Sci. (FJMS), 6(1), 83-88.

[3] . Behboodi, O. A. Karamzadeh and H. Koohy (2004). Modules Whose CerainSubmodules Are Prime, Vietnam Journal of Mathematcs, 32:3, 303-317

[4] . E. El-Bast and P. F. Smith (1988). Multiplication Modules, Comm.Alg. 16(4),755-779.

[5] . Tiras and M. Alkan. (2003) Prime Modules and Submodules, Comm.Alg.31(11), 5253-5261.



Galois Ring valued quadratic forms

M. C. Lopez-Dıaz∗ and I. F. Rua

Departamento de Matematicas, Universidad de Oviedo, C/ Calvo Sotelo s/n33007; Departamento de Matematicas, Estadıstica y Computacion, Universidadde Cantabria, Avda. de los Castros s/n 39005, [email protected]; [email protected]

2000 Mathematics Subject Classification. 11E08, 13M99

Galois Rings were first studied by W. Krull more than 80 years ago (Grun-dring) and rediscovered independently by G.J. Janusz and R. Raghavendranin the decade of the 60’s. Examples of Galois Rings include finite fields andresidual integer rings of prime power characteristic.

However, these rings have received a major attention in the last yearsdue to their nice applications to Coding Theory and Cryptography. Let usjust mention, for instance, the works [1, 2] in which Galois Rings are usedin linear presentations of non linear codes over finite fields.

In [3], the Z4-valued quadratic forms defined by E. H. Brown and studiedby J. A. Wood [4] are used to produce different families of non equivalentbinary Kerdock codes. An invariant was included by E.H. Brown along withthe definition of Z4-valued quadratic forms. J.A. Wood proved later thatthis invariant classifies, together with the type of the corresponding bilinearform (alternating or not), nonsingular Z4-valued quadratic forms.

We consider quadratic forms that take values in a Galois Ring of arbi-trary characteristic and show that their study can be reduced to the case ofcharacteristic p2. In this case we introduce and study the main properties ofan invariant, that is considered in a Galois Ring of characteristic p3. This in-variant classifies, together with the type of the corresponding bilinear form(alternating or not), nonsingular Galois Ring valued quadratic forms. More-over, simple proofs of Witt’s Cancelation and Extension Theorems followfrom the study of this invariant.

This remedy the limitations in the exposition of Wood’s results and canbe applied to the quadratic forms taking values in a Galois Ring of charac-teristic 4 used to construct different families of non necessarily equivalentGeneralized Kerdock codes over a finite field of characteristic 2 ([5]).

References

[1] Nechaev, A. A., Kerdock’s code in cyclic form, Diskret. Mat. 1 (4) (1989),123–139.



[2] Hammons Jr, A., Kumar, P. V., Calderbank, A. R., Sloane, N. J. A., Sole,P., The Z4-linearity of Kerdock, Preparata, Goethals, and related codes, IEEETrans. Inform. Theory 40 (2) (1994), 301–319.

[3] Calderbank, A. R., Cameron, P. J., Kantor, W. M., Seidel, J. J., Z4-Kerdockcodes, orthogonal spreads, and extremal Euclidean line-sets, Proc. LondonMath. Soc. 75 (1997), 436–480.

[4] Wood, J. A., Witt’s extension theorem for mod Four Valued Quadratic Forms,Trans. Amer. Math. Soc. 336 (1) (1993), 445–461.

[5] Gonzalez, S., Martınez, C., Rua, I. F., Symplecic Spread based GeneralizedKerdock Codes, Submitted (2005).



Torsion-free modules of projective dimension one

Jose Marıa Lopez*, Agustın Marcelo, Felix Marcelo and Cesar Rodrıguez

Dpto. de Matematicas, Universidad de Las Palmas de Gran Canaria, 35017 LasPalmas de Gran Canaria, [email protected]

2000 Mathematics Subject Classification. Primary 13C05; Secondary 13A02,13B25, 13E05

Let R be a Noetherian local domain and let F π−→ M −→ 0 be a minimalepimorphism of a torsion-free finitely generated and non-free R-module M,where F is a free R-module of finite rank, wich is completely determined byM up to an isomorphism. Dualizing the short exact sequence 0 −→ N

ϕ−→F

π−→ M −→ 0,we obtain 0 −→ M∗ π∗−→ F ∗ ϕ∗−→ N∗ −→ 0. The ‘codual

module’ of M is defined to be the R-module cd(M) = Imϕ∗, which is alsoa torsion-free finitely generated and non-free R-module. In this work weshow that ‘taking the codual module’ is a dualizing functor and this fact isthen used to obtain a classification of the torsion-free finitely generated andnon-free modules over a regular local ring by means of H1

a (cdM), where a

is the corresponding radical ideal of the non-free locus of cd(M).

References

[1] Bruns, W., Herzog, J., Cohen-Macaulay Rings. Cambridge Univ. Press, Cam-bridge, 1993.

[2] Eisenbud, D., Conmutative Algebra. With a view toward algebraic Geometry.Sringer-Verlag, New York, 1995.

[3] Matsumura, H., Conmutative Rings Theory. Cambridge Univ. Press, Cam-bridge, 1986.

[4] Simis, A., Ulrich, B., Vasconcelos, W., Rees algebras of modules, Proc. LondonMath. Soc.87 (2003), 610– 646.



Twisted Tensor Products in Noncommutative Geometry

Javier Lopez Pena

Algebra Department, University of Granada, Avda. Fuentenueva s/n E-18071,Granada, [email protected]

2000 Mathematics Subject Classification. 16S35; 16S38; 16S40; 16S80; 16W30;57T05; 58B32; 81T75

The difficulty of constructing concrete, nontrivial examples of noncommu-tative spaces starting from simpler ones is a common problem in all differentdescriptions of noncommutative geometry. If we think of the commutativesituation, we have an easy procedure, the cartesian product, which allowsus to generate spaces of dimension as big as we want from lower dimen-sional spaces. Thinking in terms of the existing dualities between the cate-gories of spaces and the categories of (commutative) algebras, the naturalreplacement for the cartesian product of commutative spaces turns out tobe the tensor product of commutative algebras. The tensor product has of-ten been considered a replacement for the product of spaces represented bynoncommutative algebras. Following some ideas of [1], we propose a “non-commutative” replacement of the tensor product of two algebras, which fitsbetter as an analogue of the product of two noncommutative spaces andin the same time turns out to be a useful tool for overcoming the lack ofexamples formerly mentioned. As a first step towards our aim of buildinggeometrical invariants over these structures, we will show how to build thealgebras of differential forms and how to lift the involutions of ∗–algebrasto the iterated twisted tensor products.

References

[1] A. Cap, H. Schichl, and J. Vanzura. On twisted tensor products of alge-bras. Comm. Algebra, 23:4701–4735, 1995.

[2] P. Jara Martınez, J. Lopez Pena, F. Panaite, F. Van Oys-taeyen. On iterated twisted tensor products of algebras. Preprint,arXiv:math.QA/0511280.

[3] J. Lopez Pena, Twisted Tensor Products in Noncommutative Geometry.Master Thesis, University of Antwerp, 2006.



Discussing parametric Grobner bases: on the canonicity ofDisPGB 4.1

Montserrat Manubens*, Antonio Montes

Dept. Matematica Aplicada 2, Universitat Politecnica de Catalunya, JordiGirona, 1-3 E-08034 Barcelona, [email protected], [email protected]

2000 Mathematics Subject Classification. 13P

Parametric polynomial systems have been studied from many points ofview. There are various methods devoted to study the behavior of thesesystems depending on the values of the parameters. Among others, we em-phasize dynamic evaluation [1], Alternative Comprehensive Grobner bases[4], DisPGB algorithm [2, 3] and Canonical Comprehensive Grobner bases[5].

DisPGB algorithm performs a dichotomic discussion over the values ofthe parameters in a binary tree, obtaining a partition of the parameter spaceand, for each class of the partition, a Grobner basis of the ideal generated bythe specialized polynomials. All previous releases of this algorithm were non-canonical, meaning that the results reached with DisPGB were algorithm-depending.

In this latest release of DisPGB we have implemented a recursive gener-alization of the idea of discriminant ideal defined in [2], determining, for eachgiven system, a finite increasing chain of ideals called discriminant chain,a sequence of nested varieties associated to the system and, moreover, acanonical description for most of the classes of parameter specializations.

We will present these new results over the improvements we have madeon the latest release of DisPGB, with some clarifying examples, outstandingthe cases whose canonical character can now be stated.

References

[1] Della Dora, J., Dicrescenzo, C., Duval, D., About a new method of computingin algebraic number fields. In EUROCAL’85 (ed. by G. Goos, J. Hartmanis).Springer LNCS 204, Vol 2 (1985), 289–290.

[2] Manubens, M., Montes, A., Improving DISPGB Algorithm Using the Discrimi-nant Ideal. Proceedings of Algorithmic Algebra and Logic 2005 (2005), 159–166.Also to be published in Jour. Symb. Comp.

[3] Montes, A., New Algorithm for Discussing Grobner Bases with Parameters.Jour. Symb. Comp., 33(1-2) (2002), 183–208.



[4] Suzuki, A., Sato, Y., An alternative approach to comprehensive Grobner bases.Proceedings ISSAC 2002. ACM-Press (2002), 255–261.

[5] Weispfenning, V., Canonical Comprehensive Grobner Bases. Jour. Symb.Comp. 36 (2003), 669–683.



Subgroups embedded in direct products

B. Brewster, A. Martınez-Pastor* and M. D. Perez-Ramos

Escuela Tecnica Superior de Informatica Aplicada, Departamento de MatematicaAplicada e IMPA-UPV, Universidad Politecnica de Valencia, Camino de Vera,s/n, 46022 Valencia, Spain

2000 Mathematics Subject Classification. 20D

In this poster we present an up-to-date account of recent achievementsregarding embedding properties in direct products of finite groups. Our aimis to understand how subgroups with various embedding properties can bedetected and characterized in the subgroup lattice of a direct product oftwo groups in terms of the subgroup lattices of the two groups.

Normal and subnormal subgroups in direct products had been studiedpreviously. J. Evan started in 2001 the study of permutable subgroups indirect products. Since permutability is an embedding property which liesbetween normality and subnormality, his first aim was to find characteri-zations of permutable subgroups of direct products like those for normalityand subnormality. This was nicely achieved for diagonal-type subgroups(i.e. subgroups of a direct product with trivial intersection with the com-ponents) in [3]. Next Evan obtained a necessary and suficient condition foran arbitrary subgroup of a direct product to be permutable ([4]).

The following embedding property under consideration in this contextwas the cover-avoidance property. This study was carried out by J. Petrilloin [5].

The next contribution in this area has been the characterization of nor-mally embedded subgroups in direct products, due to the authors ([1]).This property is also an extension of normality and plays an importantrole mainly in the study of soluble groups. System permutable subgroupsin direct products of soluble groups are also characterized in [1].

Finally, we will report about a current research concerning pronormalityand local pronormality which is being carried out by the authors ([2]).

References

[1] B Brewster, A. Martınez-Pastor and M.D. Perez-Ramos. Normally embeddedsubgroups in direct products of groups. To appear in J. of Group Theory.

[2] B Brewster, A. Martınez-Pastor and M.D. Perez-Ramos. Pronormal subgroupsin direct products of groups. Preprint.



[3] J. Evan. Permutable diagonal-type subgroups of G×H. Glasgow Math. J. 45(2003), 73-77.

[4] J. Evan. Permutable subgroups of a direct product. J. Algebra 265 (2003),734-743.

[5] J. Petrillo. The Cover-Avoidance Property in Finite Groups. Dissertation -Binghamton University, May 2003.



The 1-type of a Waldhausen K-theory spectrum

Fernando Muro, Andrew Tonks

Max-Planck-Institut fur Mathematik, Vivatsgasse 7, 53111 Bonn, Germany;London Metropolitan University, 166–220 Holloway Road, London N7 8DB, [email protected]; [email protected]

2000 Mathematics Subject Classification. 19B99, 16E20, 18G50, 18G55

In this poster we show an algebraic model D∗C for the 1-type P1KC of theK-theory spectrumKC of a Waldhausen category C in the sense of [Wal85].The object D∗C is a stable quadratic module in the sense of [Bau91]. This isa very special kind of strict symmetric monoidal category where all objectsare strictly invertible with respect to the monoidal structure.

The model D∗C consists of a diagram of groups

(D0C)ab ⊗ (D0C)ab 〈−,−〉−→ D1C∂−→ D0C,

satisfying certain axioms, such that there is an exact sequence

0 → K1C → D1C∂−→ D0C → K0C → 0.

The object D∗C is defined by a presentation in terms of generators andrelations in the spirit of Nenashev, who gave a model for K1 of an exactcategory in [Nen98].

The important features of our model are the following:

• It is small, as it has generators given by the objects, weak equivalencesand cofiber sequences of the category C.

• It has minimal nilpotency degree, since both groups D0C and D1Chave nilpotency class 2.

• It encodes the 1-type in a functorial way, and the homotopy classesof morphisms D∗C → D∗D and P1KC → P1KD are in bijection.

References

[Bau91] H.-J. Baues, Combinatorial Homotopy and 4-Dimensional Complexes,Walter de Gruyter, Berlin, 1991.

[Wal85] F. Waldhausen, Algebraic K-theory of spaces, Algebraic and geometrictopology (New Brunswick, N.J., 1983), Lecture Notes in Math., vol. 1126,Springer, Berlin, 1985, pp. 318–419.

[Nen98] A. Nenashev, K1 by generators and relations, J. Pure Appl. Algebra 131(1998), no. 2, 195–212.



On path coalgebras of quivers with relations of tame comoduletype

Gabriel Navarro

Department of Algebra, University of Granada, Fuentenueva s/n 18071, [email protected]

2000 Mathematics Subject Classification. 18E35, 16W30

We look for an analogous result for coalgebras of the famous Gabriel’sTheorem: every basic finite dimensional algebra, over an algebraically closedfield, is the path algebra of a quiver with relations. In order to find this result,we take into consideration the concept of path coalgebra of a quiver withrelations in the sense of Simson, see [5], and show that not every coalgebrais of this kind. Examples and also a criterion to decide whether or not acoalgebra is the path coalgebra of a quiver with relations may be found, see[2]. Nevertheless, the counterexamples we found are of wild comodule typeand, furthermore, a coalgebra with such property seems close to be wild.This suggests that the result could be true for tame coalgebras. The key-tool for this purpose is the use of the theory of localization, as described byGabriel in [1], in the context of the coalgebra theory, see [4]. In particular, werelate the tameness and wildness of a coalgebra and its localized coalgebrasin order to prove the following result [3]: Let K be an algebraically closedfield and Q be an acyclic quiver. Then any tame admissible subcoalgebra ofKQ is the path coalgebra of a quiver with relations.

References

[1] P. Gabriel, Des categories abeliennes, Bull. Soc. Math. France, 1962, 90, 323-448.

[2] P. Jara, L. M. Merino and G. Navarro, On path coalgebras of quivers withrelations, Colloq. Math. 102 (2005), 49-65.

[3] P. Jara, L. M. Merino and G. Navarro, Localization of tame and wild coalgebras,preprint, 22 pages, 2006.

[4] P. Jara, L. M. Merino, G. Navarro and F. Ruiz, Localization in coalgebras.Stable localizations and path coalgebras, Comm. Algebra, to appear, 2006.

[5] D. Simson, Coalgebras, comodules, pseudocompact algebras and tame comod-ule type, Colloq. Math., 90 (2001), 101-150.



On products of finite groups

Adolfo Ballester-Bolinches and M. C. Pedraza-Aguilera*

Departament d’Algebra, Universitat de Valencia, Dr. Moliner 50, 46100Burjassot, Valencia, Spain; Departamento de Matematica Aplicada, Instituto deMatematica Pura y Aplicada, ETS de Informatica Aplicada, UniversidadPolitecnica de Valencia, 46022 Valencia, [email protected]; [email protected]

2000 Mathematics Subject Classification. 20D10, 20F17, 20D40

Factorized groups G = AB have played an important role in the theory ofgroups in the last fifty years. Their structure is restricted by its subgroups Aand B. Well-kown results in this line are the Ito’s Theorem or the Theoremof Kegel–Wielandt about the solubility of the product of two nilpotentfactors. The background of these results is: What can we say about G = ABif special properties of A and B are known? Some special cases are those ofdirect or central products. To create intermediate situations, it is usual toconsider products of groups whose factors are linked by certain relations.A seminal paper is the one by Asaad and Shaalan [1] about totally andmutually permutable products of supersoluble groups. Later weaker versionsas permutability of each factor only with some families of subgroups of theother one (subnormal, maximal subgroups) have been analyzed. This resultswere the beginning of an intensive productive line of research in this area,even in the infinite case (see [2] and [3]). We present now some recentnew results in this context. On one hand we study mutually permutableproducts of SC-groups [4], and on the other hand we analyze the structureof mutually permutable products of nilpotent groups [5] obtaining that inthis case a group of odd order, is abelian by nilpotent.

Our aim is therefore to provide an overall knowledge of the backgroundand recent work on products of groups.

References

[1] M. Asaad and A. Shaalan, On the supersolvability of finite groups. Arch. Math.53 (1989), 318-326.

[2] A. Ballester-Bolinches, M. D. Perez-Ramos, M. C. Pedraza-Aguilera, Totallyand mutually permutable products of finite groups. Groups St. Andrews 1997 inBath I, 65-68. London Math. Soc. Lecture Note Ser., 260. Cambridge UniversityPress, Cambridge, 1999.



[3] J.C. Beidleman, H.A. Heineken, Survey of mutually and totally permutableproducts in infinite groups. Topics in infinite groups, Topics in infinite groups,45-62, Quad. Mat., 8, Dept. Math., Seconda Univ. Napoli, Caserta, 2001.

[4] A. Ballester-Bolinches, John Cossey and M. C. Pedraza-Aguilera, On mutuallypermutable products of finite groups, J. Algebra 294 (2005), 127-135.

[5] A. Ballester-Bolinches, J. C. Beidleman, John Cossey, H Heineken and M.C. Pedraza-Aguilera, Mutually permutable products of nilpotent groups. Toappear in Rend. Sem. Mat. Padova.



From finite to locally finite groups

Adolfo Ballester-Bolinches and Tatiana Pedraza*

Departament d’Algebra, Universitat de Valencia, Dr. Moliner 50, 46100Burjassot, Valencia, Spain; Departamento de Matematica Aplicada, Instituto deMatematica Pura y Aplicada, ETS de Informatica Aplicada, UniversidadPolitecnica de Valencia, 46022 Valencia, [email protected]; [email protected]

2000 Mathematics Subject Classification. 20F50, 20F16, 20F19, 20E15

A group is said to be locally finite if every finite subset of G generates a finitesubgroup. The class of locally finite groups is placed near the cross-roads offinite group theory and the general theory of infinite groups. Some theoremsabout finite groups can be phrased in such a way that their statements stillmake sense for locally finite groups. However, in general, Sylow’s Theoremsdo not hold in the class of locally finite groups and there are a number ofgeneric examples which show that locally finite groups can be very variedand complex. If we restrict our attention to locally finite groups that areradical and satisfy the minimal condition on p-subgroups for every primep, then the Sylow p-subgroups are very well behaved and this behavior hasguaranteed the successful development of formation theory in this class ofgroups.

Therefore it is appropriate to study this class of locally finite groups,denoted by cL, in more detail. In this work, we will show that this universeof groups is a suitable context to generalize nilpotency and supersolubilityin such a way that their properties in the finite case are preserved. Of courseminor modifications have to be made to allow for the lack of finiteness. Thuswe extend to cL some well-known facts from the theory of finite solublegroups ([1, 2, 3]). Moreover, we investigate cL-groups in which normalityis a transitive relation obtaining some characterizations similar to those offinite soluble groups ([4]).

References

[1] Ballester-Bolinches, A. and Pedraza, T., On a class of generalized nilpotentgroups, J. Algebra 248 (2002), 219–229.

[2] Ballester-Bolinches, A. and Pedraza, T., On products of generalized nilpotentgroups. Forum Math. 16(5) (2004), 717–724.

[3] Ballester-Bolinches, A. and Pedraza, T., A class of generalized supersolublegroups, Publ. Mat., Barc. 49 (2005), 213–223.



[4] Ballester-Bolinches, A., Heineken, H. and Pedraza, T., On a class of locallyfinite T -groups , Forum Math., to appear.

[5] Dixon, M.R., Sylow Theory, Formations and Fitting classes in Locally FiniteGroups. Series in Algebra 2, World Scientific, River Edge, NJ, 1994.



Super-characters of finite algebra groups over finite rings

Carlos A. M. Andre, Ana Margarida Neto, Alejandro Nicolas*

Departamento de Matematica e Centro de Estruturas Lineares e Combinatorias,Faculdade de Ciencias da Universidade de Lisboa, Rua Ernesto de Vasconcelos,Bloco C1, Piso 3, 1700 Lisboa, Portugal; Instituto Superior de Economia eGestao, Universidade Tecnica de Lisboa, Rua do Quelhas 6, 1200-781 Lisboa,Portugal e Centro de Estruturas Lineares e Combinatorias, Av. Prof. GamaPinto 2, 1649-003 Lisboa, Portugal; Departamento de Matematicas, Facultad deCiencias, Universidad de Oviedo, Calvo Sotelo s/n, 33007 Oviedo, [email protected]; [email protected];[email protected]

2000 Mathematics Subject Classification. 20C15, 20G40

Let O be a discrete valuation ring of characteristic zero, let p ∈ O be anyprime element and let m = pO be the (unique) maximal ideal of O. LetA be a finite-dimensional algebra over O which is a finitely generated freeO-module. If the Jacobson radical is J = J(A), then the set G = 1 + J isa subgroup of the units of A called algebra group.

For any positive integer r ∈ N, we consider the two-sided ideal prJ ofJ and the subgroup Gr = 1 + prJ of G. Since prJ is an ideal of J , Gr isnormal in G and the quotient G/Gr is a finite group.

In this work we construct the super-characters of the finite group G/Gr

and we study their main properties. Finally, we give some examples.

References

[1] C. A. M. Andre, Basic Characters of the Unitriangular Group, Journal of Al-gebra, 175 (1995), 287-319.

[2] C. A. M. Andre, Basic Characters of the Unitriangular Group (for arbitraryprimes), Proceedings of the American Mathematical Society, 130 (2002), 1943-1954.

[3] Ery Arias-Castro, Persi Diaconis, Richard Stanley, A super-class walk on upper-triangular matrices, Journal of Algebra, 278 (2004), 739-765.

[4] Ning Yan, ”Representation Theory of the Finite Unipotent Linear Groups”,Ph. D. Thesis, University of Pennsylvania, 2001.



Relative Brauer groups of rational function fields of genus 1curves over local nondyadic fields Qp

A. V. Prokopchuk

Department of Algebra, Institute of Mathematics, National Academy of Sciencesof Belarus, 11 Surganov Str., Minsk, 220072, [email protected]

2000 Mathematics Subject Classification. 16K50

Let K/F be a field extension and ResK/F the natural homomorphism ofBrauer groups BrK, BrF induced by the inclusion K ↪→ F . The kernelkerResK/F of ResK/F is considered in case F = Qp, p is odd and K is thefunction field of curves over F defined by an equation y2 = ax4 + b in thecase F = Qp, p is odd are considered. The general situation can be reducedto the case where a ∈ u, p, up, u is unit in Zp, and 0 6 v(b) 6 3 for thecanonical valuation v of Zp. For the above mentioned curves the criterionof non-triviality of kerResK/F is given. In special case it looks as follows.

Theorem. Let C/Qp be defined by the equation C : y2 = px4+b, b ∈ Q∗p.

a) If v(b) = 0, then |Br(Qp(C)/Qp)| = 2 ⇔ b is not square.b) If v(b) = 1, then |Br(Qp(C)/Qp)| = 2 ⇔ p ≡ 3( mod 8), p−1b is not

square or square but not fourth power for p ≡ 1( mod 8) and p−1b is notsquare or fourth power for p ≡ 5( mod 8).

c) If v(b) = 2, then |Br(Qp(C)/Qp)| = 2 ⇔ p−2b is not square.c) If v(b) = 3, then |Br(Qp(C)/Qp)| = 2.Remark The case, where F = R or F = Q2, were discussed in [1].

References

[1] A.V. Prokopchuk, V.I. Yanchevskii, Relative Brauer groups of rational functionfields of genus 1 curves over special fields, Tr. Inst. Math. 13. (2005) 60 - 73.



Local cohomology with supports in the non-free locus

Jose Marıa Lopez, Agustın Marcelo, Felix Marcelo and Cesar Rodrıguez*

Dpto. de Matematicas, Universidad de Las Palmas de Gran Canaria, 35017 LasPalmas de Gran Canaria, [email protected]

2000 Mathematics Subject Classification.

Let R be a Noetherian ring and let M be a finitely generated R-module.As is well known, the set of points p ∈SpecR such that Mp is a free Rp-module, is an open subset in the Zariski topology. Hence, its complementC is a closed subset -called the non-free locus of M - whose correspondingradical ideal is denoted by a = a(M) = =(C). In this work the groups oflocal cohomology with supports in the non-free locus of a module are used inorder to obtain classifications of two classes of modules. More precisely, weprove that a reflexive finitely generated module M over a regular local ringwhose dual module is of projective dimension one is completely determinedby H2

a (M). Similarly, we obtain a classification of the ideal modules over alocal regular ring by means of H1

a (M).

References

[1] Bourbaki, N., Elements de Mathematique, Algebre Commutative. Hermann,Paris, 1961.

[2] Bruns, W., Herzog, J., Cohen-Macaulay Rings. Cambridge Univ. Press, Cam-bridge, 1993.

[3] Matsumura, H., Conmutative Rings Theory. Cambridge Univ. Press, Cam-bridge, 1986.

[4] Simis, A., Ulrich, B., Vasconcelos, W., Rees algebras of modules, Proc. LondonMath. Soc.87 (2003), 610– 646.



Hyper-operations as a tool for science and engineering

Konstantin A. Rubtsov*, Giovanni F. Romerio

Center of Engineering of New Information Processing, Belgorod Shukhov StateTechnological University (BSTU), Russian Federation; Independent Consultant,Via Torino 48, 12037 Saluzzo (CN), [email protected]; 20k30, 03D99, 11A25, 11H99

2000 Mathematics Subject Classification. 20k30, 03D99, 11A25, 11H99

New mathematical tools for describing discontinuous signals such as stepfunctions, infinite sampling signals, or abstract concept like “dot events” areproposed, by introducing a new arithmetical operation called “zeration”. Anew number notation is also proposed by using an iterated exponentiationoperation, known as ”tetration”. Both objectives are justified by Acker-mann’s Function, from which an infinite series of hyper-operation can bedefined.

In particular, the new zeration operation immediately follows, with arank lower than addition showing very simple and clear properties. Its in-verse operation (deltation) generates new numbers (delta numbers), theproperties of which are being analyzed. It can be proved that −(a∧b) =a∧(4b), i.e. that delta numbers are a new class of numbers, which can beobtained as logarithms of negative numbers (traditionally represented bymulti-value complex numbers). Moreover, zeration allows a new approachto the non-standard ”actual infinite”.

The study of tetration (super-power, tower) and its two inverse opera-tions (super-root and super-logarithm), shows interconnections with num-bers obtained from zeration and other inverse hyper-operations with rankslower than addition.

A new number notation format is also proposed, formally similar tothe “floating point” format, but using tetration instead of exponentiation.With this format, it is easy to represent very large numbers, as well asthose obtained by inverse operations with ranks lower than addition. Apractical machine storage format has been implemented. Prototypes of ahyper-calculator and of a notation hyper-converter have also been devel-oped.

References

[1] Rubtsov, C. A., Romerio, G. F., Ackermann’s Function and New ArithmeticalOperations. Manuscript cited in the bibliography of Stephen Wolfram’s ”A New



Kind of Science”, 2003.See also: http://www.rotarysaluzzo.it/filePDF...zioni%20(1).pdf

[2] Rubtsov, C. A., New mathematical objects. BelGTASM, Belgorod, Russia;NPP-Informavtosim, Kiev, Ukraine, 1996, Monograph, 251 p. (In Russian).See also: http://numbers.newmail.ru/pdf/book_rus.pdf

[3] Rubtsov, C. A., Algorithms ingredients in a set of algebraic operations. Cyber-netics 3 (1989), 111-112. (In Russian).



Lie quotients for skew Lie algebras

Miguel Cabrera, Juana Sanchez Ortega*

Dpto. de Algebra, Geometrıa y Topologıa, Universidad de Malaga, 29071 Malaga,[email protected]

2000 Mathematics Subject Classification.

Algebras of quotients of associative algebras have played a pivotal role inthe development of the theory, since their structural properties give a betterinsight of important classes of rings. Moreover, they are necessary in orderto properly introduce generalized identities.

The notion of “being an algebra of quotients” has found recently pa-rallels for non-associative algebras, both in the Jordan and the Lie settings(see e.g. [3, 2, 1] and [5, 4]). In this work we shall be concerned with theconcept of algebra of quotients for Lie algebras, as introduced and exploredin [5].

In order to foresee the importance of this concept in the non-associativesetting, F. Perera and M. Siles Molina undertook in [4] a study of therelationship between the Lie and associative quotients, which includes theanalysis of dense subalgebras of Lie algebras.

Let A be a semiprime associative algebra with an involution ∗ over afield of characteristic not 2, let KA be the Lie algebra of all skew elementsof A, and let ZKA

denote the annihilator of KA. We prove that if Q isa ∗-subalgebra of Qs(A) (the Martindale symmetric algebra of quotientsof A) containing A, then KA/ZKA

is a dense subalgebra of KQ/ZKQand

KQ/ZKQis a Lie algebra of quotients of KA/ZKA

. The similar result holdsfor the extension [KA,KA]/Z[KA,KA] ⊆ [KQ,KQ]/Z[KQ,KQ].

References

[1] Anquela, J. A., Gomez Lozano, M., Garcıa, E., Maximal algebras of Martindale-like quotients of strongly prime linear Jordan algebras, J. Algebra 280 (2004),367-383.

[2] Martınez, C., The ring of fractions of a Jordan algebra, J. Algebra 237 (2001),798-812.

[3] McCrimmon, K., Martindale systems of symmetric quotients. Algebras GroupsGeom. 6 (1989), 153-237.

[4] Perera, F., Siles Molina, M., Associative and Lie algebras of quotients. Preprint,2005.



[5] Siles Molina, M., Algebras of quotients of Lie algebras. J. Pure Appl. Algebra188 (2004), 175-188.



Structures derived from paragraded groups

Mirjana Vukovic

Department of Mathematics, University of Sarajevo, Bosnia and [email protected]

2000 Mathematics Subject Classification. 08A05, 16W50, 20L05, 16Y60

In our common papers [2] – [4] and monograph [5] M. Krasner and myselffirst introduced the notion of an para- and extra- graduation, as well as thecorresponding groups called para- and extragraded which is at the sametime a generalization of classical graduation, as defined by Bourbaki, andan extension of the earlier work done by M. Krasner [1].

Here, we first observe structures derived from paragraded groups. Westart with homogeneous groups and prouve that from para- (extra-) grad-uation of the group G we get corresponding graduation of its homogeneoussubgroup g, and how it can be characterized homogeneous part h of thehomogeneous group g, in other words subgroupoid h of the extragroupoidH.

Further, we define factor structure H/h and for that structure we provethe first and the second theorem of isomorph. With the help of quasiho-mogeneous isomorphysm Φ paragraded group (G,·, E) to paragraded group(G’,·, E’) it is defined paragraded group (G,·, E (Φ)), so-called aglutinat of(G,·, E), which is paragraded group, if it is (G’,·, E’).

For the family of the paragraded groups it is defined, in the natural way,the direct product and the direct sum. Those are, again paragraded groups,and the direct sum is homogeneous subgroup of the direct product. Thedirect product (direct sum) of the paragraded groups is extragraded group,iff all of the factors are extragraded groups. However, the direct product ofthe graded groups is only extragraded group, except if at most one of thefactors with non-trivial graduation, that is ∆α 6= {0}.

References

[1] Krasner, M., Anneaux gradues generqux, Colloque d’algebre, Univ. Rennes 1(1980), 209-308.

[2] Krasner, M., Vukovic, M., Structures paragraduees, groupes, anneaux,modulesI, Proc. Japan Acad. 62 (1986), Ser.A, No. 9, 350-352.

[3] ——–, II, Proc. Japan Acad. 62 (1986), Ser.A, No.10, 389-391.



[4] ——–, Structures paragraduees (groupes, anneaux, modules), Queen’s Pa-pers inpure and Applied Mathematics, No. 77, Queen’s University, Kingston,Canada (1987), p.163.

[5] Vukovic, M., Structures graduees et paragraduees, Prepublications del’Universite Grenoble I, No. 536 (2001), 1- 40. http://www-fourier.ujf-grenoble.fr/prepublications.html


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