SECOND HONOLULU CONFERENCE - University of Hawaiiadolf/Meeting/abstracts.pdf · 2001-09-24 · SECOND HONOLULU CONFERENCE ABSTRACTS ... In this paper we show an extended version of

SECOND HONOLULU CONFERENCE

ABSTRACTS

Extending the notion of cotorsion abelian groupsto modules over commutative domains

Silvana Bazzoni, University of Padova

ABSTRACT. The notion of cotorsion abelian group, introduced by Harrison in1959, was generalized in various different ways for modules over any associative ringwith unit. If R is a commutative domain, an R-module C is called weakly cotorsion(or Matlis cotorsion) if Ext1R(Q, C) = 0, where Q denotes the field of quotients ofR. C is called strongly cotorsion (or Warfield cotorsion) if Ext1R(M, C) = 0 forevery torsion-free R-module M . If R is any associative ring with unit, a module Cis called cotorsion (or Enochs’ cotorsion) if Ext1R(F, C) = 0 for any flat R-moduleF . The three notions of weakly cotorsion, cotorsion and strongly cotorsion modulesconcide if and only if R is a Dedekind domain. The notions of cotorsion and stronglycotorsion modules coincide if and only if R is a Prufer domain. We characterize thecommutative domains R for which the class of Matlis cotorsion modules coincideswith the class of Enoch cotorsion modules. We prove that they are exactly thealmost perfect domains, i.e., they have the property that R/I is a perfect ring forevery non-zero ideal I of R. For almost perfect domains we prove a result analogousto Bass’ Theorem on perfect rings.

Silvana Bazzoni email: [email protected]. di Mat. Pura e Appl. phone: +39-049-8275947Via Belzoni 7 fax: +39-049-827589235131 Padova, Italy

Forcing and independence results

Paul Cohen, Stanford University

Paul Cohen email: [email protected] of MathematicsStanford UniversityPalo Alto, CA 94305

1

2 ABSTRACTS

Functorial topologies can measure algebraic invariantsof the abelian groups

Dikran Dikranjan, Universita di Udine

ABSTRACT. For a functorial topology T we say that two abelian groups G andH are T -homeomorphic, if the topological spaces (G, TG) and (H, TH) are home-omorphic. Clearly, T -homeomorphic groups must have the same size, we discusspreservation of some algebraic invariants (e.g., free rank, p-rank, etc.) under T -homeomorphisms, where T is the initial topology of Hom(−,T) and T = R/Z isthe circle group. Here is a sample result:

Theorem. Let p be a prime and let κ be a cardinal greater than i2p−1. If thepowers Gκ and Hκ of two abelian groups G and H are T -homeomorphic, thenrp(G) is finite iff rp(H) is finite.

The proof of the theorem relies on the following “straightening law” that inverts,to a certain extent, the well known fact that all homomorphisms are T -continous.If f : G =

⊕κ Z(p)→ H is a T -continuous map with f(0) = 0 and κ > i2p−1, then

f coincides on a large chunk of G with the restriction of a homomorphism G→ H .

Dikran Dikranjan email: [email protected]. di Mat. Pura e Appl.Universita di UdineUdine, Italy

Tight subgroups of almost completely decomposable groups

Ulrich Dittmann, Universitat Wurzburg

ABSTRACT. In this paper we show an extended version of the theorem of Be-zout, give a new criterion for the tightness of a completely decomposable subgroup,derive some conditions under which a tight subgroup is regulating and generalizea theorem of Campagna. We give examples of almost completely decomposablegroups, all of whose regulating subgroups do not have a quotient with minimalexponent.

Ulrich Dittmann email: [email protected] Wuerzburg phone: +49 931 8885016Mathematisches Institut fax: +49 931 8884611Am Hubland97074 Wuerzburg, Germany

Completely decomposable groups with one distinguished cd subgroup

Manfred Dugas, Baylor UniversityK.M. Rangaswamy, University of Colorado, Colorado Springs

ABSTRACT. We define a category CD(T, p), p a prime and T a set of types,consisting of all pairs V = (C, D), where C is a completely decomposable groupwith critical type set T and D a completely decomposable subgroup of C such that

SECOND HONOLULU CONFERENCE 3

peC is contained in D. We show that while indecomposables in this category haverank 1 if T is an antichain, we observe “wild” behavior if T is not an antichain, i.e.,T has comparable elements.

Manfred Dugas email: [email protected] of Mathematics phone: (254)-710-4886Baylor University department phone: (254)-710-3561Waco, TX 76798-7328 fax: (254)-710-3569

The affinity of set theory and abelian group theory

Paul Eklof, University of California, Irvine

Paul Eklof email: [email protected] of Mathematics office phone: 949-824-6595University of California Irvine fax: 949-824-7993Irvine, CA 92697-3875 http://www.math.uci.edu/faculty/peklof.html

On covers and covering morphisms

Robert El Bashir, Charles University Prague

ABSTRACT. We consider covers of modules with respect to full subcategoriesof module categories. For a given class of modules F such that F–covers exist,we look at extensions of modules which have the same F–cover, so called coveringmorphisms. We give a characterization of covering morphisms and focus on thecase when F is the class of flat modules.

Robert El Bashir email: [email protected] Fyzikalni FakultaUniversita KarlovaSokolovska 83Praha 8, Karlin, Czech Republic

Abelian Groups in Russia

Quotient divisible mixed groups

Alexander Fomin, La Universidad Antonio Narino

Alexander Fomin email: [email protected] Universidad Antonio Narino fax: (571) 221-5177Olimpiadas Colombianas en MatematicasCra 38 \# 58 A 77Bogota, Colombia

4 ABSTRACTS

Abelian groups in Hungary

Additive ideal theory in non–noetherian domains

Laszlo Fuchs, Tulane University

Laszlo Fuchs email: [email protected] of Mathematics phone: 504-865-5727Tulane UniversityNew Orleans, LA 70118-5698

Abelian Groups in England and Germany

Rudiger Gobel, Universitat Essen

********************************************************************* portable phone: [+49] (0)179-4790228 ********************************************************************** Universitaet Essen | Tel.: [+49] (0)201/ 183-2410 ** FB 6 | Fax.: [+49] (0)201/ 183-3219 ** Mathematik und Informatik | Home(Phone|FAX):[+49](0) 201/484848 ** D 45117 Essen | EMAIL: [email protected] ** HOME PAGE: http://www.uni-essen.de/~mat101 *********************************************************************

Some aspects of minimality in Abelian Groups

Brendan Goldsmith, Dublin Institute of Technology

Brendan Goldsmith email: [email protected] of MathematicsDublin Institute of TechnologyKevin StreetDublin 8, EireIreland

On spectra of graphs related to free products of abelian groups

Vishal Goundar, University of the South Pacific

ABSTRACT. We will discuss an algebraic approach to the construction of ex-panders close to Ramanujan graphs using the free product F (G) of two copies ofthe finite group G. The free product G ? G of two copies of a finite group G canbe used to define family of graphs Γi(G) which are quotients of the tree related toF (G). For some families from this class the second largest eigenvalue is boundedby 2

√|G| (see [1])

It is known that family Γi, i = 1, 2, . . ., of k–regular graphs is a family of ex-panders (see [2]), if the second largest eigenvalue is bounded away from k.

Some families of graphs Γi(G) can be defined in terms of equations over finitefield Fq. It allow us to compute spectra of them efficiently.

We will present the results of computer simulations. In some cases we get clearpatterns and conjecture the close formulas for the second largest eigenvalue.


References

[1] V. Ustimenko, V. Gounder, Proceedings of international conference on algebra and algebraicstructures, Sumy, Ukraine, June, 2001, 111-112.

[2] A. Lubotzky, Discrete Groups, Expanding Graphs and Invariant Measures, Progr. in Math.,Birkhauser, 1994, 125.

Vishal Goundar email: [email protected] of the South Pacific fax: 303455C/- Maths and Computing Science,P.O. Box 1168,USP, Suva, Fiji

A Note on I0-rings and I0-modulesover generalized triangular matrix rings

Hao Zhifeng, South China University of TechnologyFeng Lianggui, National University of Defence Technology

ABSTRACT. We study the I0-modules over a generalized triangular matrix rings

T =(

R M0 S

), where M is a R–S–bimodule. Our main result is to show

that a projective left T –module[(

PQ

),

(f0

)]is an I0–module if and only

if P/f(M ⊗S Q) (resp. Q) is an I0–module as left R (resp. S)–module. In par-ticular,we show that T is a I0–ring if and only if R and S are I0–rings. Moreover,Tn(R) (upper triangular matrix of R) is an I0–ring if and only if R is I0–ring.

Hao Zhifeng email: [email protected] of Applied MathematicsSouth China University of TechnologyGuangzhou 510641, P.R.ChinaDepartment of Mathematics email: [email protected] UniversityPiscataway,NJ 08854-8019,USA

Feng LiangguiDepartment of system and Engineering MathematicsNational University of Defence TechnologyChangsha 410073, P.R.China

The Kervaire-Murthy conjectures andunit-type bases in integer group rings

Ola Helenius, University of Goteborg

ABSTRACT. Let p be a semi-regular prime, let Cpn be a cyclic group of orderpn and let ζn be a primitive pn+1th root of unity. There is a short exact sequence

0→ V +n ⊕ V −

n → Pic ZCpn+1 → Cl Q(ζn ⊕ PicZCpn → 0

6 ABSTRACTS

In 1977 Kervaire and Murthy established an exact structure for V −n , proved that

Char(V +n ) ⊆ Char(V+

n ) ⊆ Cl(p)(Q(ζn−1)), where Vn is a canonical quotient of Vn,and conjectured that Char(V +

n ) ∼= (Z/pnZ)r, where r the index of irregularity of p.We prove that under a certain extra condition on p, Vn

∼= Cl(p)(Q(ζn−1)) ∼=(Z/pnZ)r and Vn

∼=⊕r

i=1(Z/pn−δiZ), where δi is 0 or 1.

Ola Helenius email: [email protected] of MathematicsChalmers University of TechnologyS-41296 GoteborgSweden

The Birth of Homological Algebra

Peter Hilton, SUNY at Binghamton

Hilton, Peter J email: [email protected] Prof Emer phone: 607-777-4867Mathematical Science DepartmentSUNY at BinghamtonBinghamton, NY 13902-6000

On determining the classification difficultyof countable torsion free abelian groups

Greg Hjorth, UCLA

Greg Hjorth email: [email protected] of MathematicsUniversity of California Los AngelesLos Angeles, CA 90095-1555

A Characterization of FGC rings

George Ivanov, Macquarie UniversityPeter Vamos, University of Exeter

ABSTRACT. Commutative rings whose finitely generated modules are directsums of cyclics are called FGC rings. These were first determined, after mucheffort, in the late 1970’s. We give a new classification which we hope may be usefulfor studying noncommutative analogues.

George Ivanov email: [email protected] of Mathematics phone: (61 2) 9850 8950Macquarie University fax: (61 2) 9850 8114NSW 2109, Australia


Local–injective and local–projective modules

Friedrich Kasch, Universitat Munchen

ABSTRACT. The notions local–injective and local–projective modules were in-spired by results about the total, which we first have to explain. Let R be a ringwith 1 ∈ R and denote by Mod–R the category of all unitary right R–modules. IfA is a small (=superfluous) submodule of M , we write A ⊆0 M ; if A is a large(=essential) submodule of M , we write A ⊆∗ M and A ⊆⊕ M means that A is adirect summand of M .

If f ∈ HomR(M, W ), then the following are equivalent (see [5]):1. There exists g ∈ HomR(W, M) such that e := gf = e2 6= 0 (e = idempotent

in End(M))2. There exists h ∈ HomR(W, M) such that d := fh = d2 6= 0 (d = idempotent

in End(W ))3. There exists 0 6= A ⊆⊕ M , B ⊆⊕ W such that the mapping A 3 a 7→ f(a) ∈

B is an isomorphism.If these properties are satisfied for f we call f partially invertible = pi (forshort). Then the total of M ,W is

Tot(M, W ) := {f ∈ HomR(M, W ) | f is not pi}This total contains the radical, the singular submodule and the cosingular sub-module of HomR(M, W ). In general, the total is a semi–ideal in Mod–R, but notadditively closed. It is a question: When is Tot(M, W ) additively closed (see [1])?

Definitions:1. The module V is called local–injective = li (for short) if and only if for every

submodule A ⊆ V which is not large in V , there exists a nonzero injectivesubmodule Q ⊆ V with A ∩Q = 0.

2. The module W is called local–projective = lp (for short) if and only iffor every submodule B ⊆W which is not small in W , there exists a nonzeroprojective direct summand P ⊆⊕ W with P ⊆ B.

With these notions we have the following theorem.

Theorem:1. For a module V the following are equivalent:

(a) For every M ∈ Mod–R, Tot(V, M) = {f ∈ HomR(V, M) | Ker(f) ⊆∗

V } = singular submodule of V, M = M(V, M)(b) V is local–injective.

2. For a module W the following are equivalent:(a) For all modules M ∈ Mod–R, Tot(M, W ) = {f ∈ HomR(M, W ) |

Im(f) ⊆0 W} = cosingular submodule of M, W = O(M, W )(b) Tot(R, W ) = O(R, W )(c) W is local–projective.

In the following we explain some further properties of li and lp modules (incom-plete list!)

Prop. 1: If V is li and has no proper large submodule, the it is a left–TOTO–module in the sense of [2]. If W is lp and has no nonzero small submodule, then it is

8 ABSTRACTS

a right TOTO–module in the sense of [2]. For modules M , W with Tot(M, W ) = 0see also [4].

Prop. 2: Direct summands of li resp. lp modules are again li resp. lp.

Prop. 3: Injective modules are li. Projective semi–perfect modules are lp. (Seealso I0–modules in [3].)

Prop. 4: If the li module V has maximum condition for injective submodules,then V is injective. If the lp module W has maximum condition for projectivedirect summands of W , then W is projective.

Prop. 5: For V the following are equivalent.1. V is li module2. V contains a direct sum U :=

⊕i∈I Qi of injective modules Qi and U ⊆∗ V .

Prop. 6: If V is li and RR is Noetherian, then V is injective.

Remark: Part 1.) of our theorem was already proved in [1], but under theassumption that V satisfies 2. in Prop. 5. The advantage here is to have theduality between li and lp and by this to obtain also part 2.) of the theorem.

References[1] Kostia I. Beidar, Friedrich Kasch, Good conditions for the Total, Proc. of

the third Korea–China–Japan Int. Symp. on Ring Theory, 1999, to appear[2] Kostia I. Beidar, Friedrich Kasch, TOTO–modules, Algebra–Berichte 76(2000),

1–11, Verlag R. Fischer, Munchen[3] H. Hamza, I0–rings and I0–modules, Math. J. Okuyama Univ. 40(1998),

91–97[4] Friedrich Kasch, Modules with zero Total, Icor 2000, Innsbruck, to appear[5] Friedrich Kasch, Wolfgang Schneider, The Total of Modules and Rings,

Algebra–Berichte 69(1992), 1–85, Verlag R. Fischer, Munchen

Friedrich Kasch email: [email protected]. 16 phone: +49-8178-549882057 Icking, Germany

Modules with the internal exchange property

Nguyen Viet Dung, Ohio University-ZanesvilleS. K. Jain and S. R. Lopez-Permouth, Ohio University-Athens

ABSTRACT. In this paper we study modules satisfying the (finite) internal ex-change property. It is shown that a right R-module M has the finite internalexchange property if and only if its endomorphism ring has the same property asa right module over itself. In the case when M is a direct sum of indecomposablemodules, then M has the (finite) internal exchange property if and only if the de-composition complements direct summands. We also show that if a module M ispure-quasi-continuous, then M has the internal exchange property if and only if ithas the finite internal exchange property.


Sergio R. Lopez-Permouth email: [email protected] of Mathematics phone: (740) 593 1262321 Morton Hall fax: (740) 593 9805Ohio University http://bing.math.ohiou.edu/~slopez/Athens, OH 45701

A note on compositions of derivations of prime rings

M. A. Chebotar, Tula State University, Tula, RUSSIAPjek-Hwee Lee, National Taiwan University, Taipei, TAIWAN

ABSTRACT. Let d1, . . . , dn be derivations of prime ring R. We will discuss whenthe composition d1 . . . dn is a nonzero derivation of R. In particular, we will answertwo questions posed by Lanski.

M. A. Chebotar email: [email protected] of Mechanics and MathematicsTula State University, Tula, RUSSIA

Pjek-Hwee Lee email: [email protected] of MathematicsNational Taiwan University, Taipei, TAIWAN

Skew derivations algebraic over prime rings and their constants

Tsiu–Kwen Lee, National Taiwan UniversityChen–Lian Chuang, National Taiwan University

ABSTRACT. Let R be a prime ring with an automorphism σ and let RF be its leftMartindale quotient ring. We first give a characterization of σ–derivations whichare left algebraic over RF modulo finite–dimensional subspaces of RF . Applyingthis characterization we prove several results concerning certain identities with σ–derivations and concerning their constants.

Chen--Lian Chuang and Tsiu--Kwen Lee email: [email protected] of Mathematics email: [email protected] Taiwan UniversityTaipei 106, TAIWAN

Purity and topological purity in locally compact abelian groups

Peter Loth, Sacred Hearts University

ABSTRACT. A closed subgroup H of a topological abelian group G is said tobe topologically pure if nH = H ∩ nG for every positive integer n. In this talk, westudy the role of purity and topological purity in the category of locally compactabelian (LCA) groups. If the torsion part of an LCA group is closed, then it is notnecessarily topologically pure. Some structural information is obtained on the LCAgroups T that are topological direct summands of every LCA group in which theyare contained as topologically pure subgroups.In case these groups T are discrete orcompact, they are completely described. We characterize those LCA groups which

10 ABSTRACTS

have no topologically pure subgroups except the trivial ones, and those LCA groupsfor which all proper closed subgroups are topologically pure.

Peter Loth email: [email protected] of Mathematics email: [email protected] Heart University office phone: 203-365-75625151 Park AvenueFairfield, CT 06432

Decomposing base–changes of B(1)–groups

Claudia Metelli, Universita di Napoli

Claudia Metelli email: [email protected]. di Matematica i Applicazioni ‘‘Renato Cacciopoli’’Universit\‘a di NapoliVia Cinthia, Complesso Monte S. Angelo, Edificio ‘‘T’’80126 Napoli, Italy

Regulator chains of Butler groups

Otto Mutzbauer, Universitat Wurzburg

ABSTRACT. For Butler groups there is a regulator as for almost completelydecomposable groups. This regulator is of finite index, hence again a Butler group.This gives rise to a regulator chain of Butler groups. All factors of this chain areinvariants of the group. There is a Butler group of rank 8 with 13 critical typeswith regulator chain of length 2. This is momentarily the minimal example of aButler group with chain length ≥ 2. Butler groups with longer regulator chain arenot known. It is conjectured that every Butler group has a finite regulator chain.

Otto Mutzbauer email: [email protected] Wuerzburg phone: +49 931 8885016Mathematisches Institut fax: +49 931 8884611Am Hubland97074 Wuerzburg, Germany

Isomorphism Classes of Uniform Groups

Michael Nahler, Universitat Wurzburg

ABSTRACT. A uniform group is a rigid almost completely decomposable groupwith a primary homocyclic regulator quotient. We count isomorphism classes ofuniform groups within a fixed near–isomorphism class. Uniform groups have rep-resenting matrices in Hermite normal form. Two representing matrices in Hermitenormal form describe isomorphic groups if and only if the rest blocks of the repre-senting matrices are typeset diagonally equivalent. We get a formula for the numberof isomorphic groups and derive an upper and lower bound for this number.


Michael Nahler email: [email protected] Wuerzburg phone: +49 931 8885016Mathematisches Institut fax: +49 931 8884611Am Hubland97074 Wuerzburg, Germany

Quasi Purifiable Subgroups and Height-Matrices

Takashi Okuyama, Toba National College of Maritime Technology

ABSTRACT. Let G be an arbitrary abelian group. A subgroup A of G is said tobe quasi purifiable in G if there exists a pure subgroup H of G containing A suchthat A is almost-dense in H and H/A is torsion. Such a subgroup H is called aquasi pure hull of A in G.

First we present a necessary and sufficient condition a torsion-free rank-one sub-group of G is quasi purifiable in G as follows.

Theorem 0.1. Let G be an abelian group and A a torsion-free rank-one subgroupof G. Then A is quasi purifiable in G if and only if, for every a ∈ A and everyp ∈ P,

hp(a) = ω implies hp(a) =∞.(0.2)

Next we use the result to compute the height-matrix of the torsion-free elementa of an abelian group G, which satisfies the condition (0.2) as follows.

Corollary 0.3. Let G be an abelian group and a ∈ G \ T . Suppose that, for ev-ery integer n = 0, either hp(pna) < ω or hp(pna) = ∞. Let m = hp(a) and let{tn} be the p-overhang set of 〈a〉 in G. Define cn = max{hG/〈a〉

p (y + 〈a〉) | y ∈〈a〉tn

G (p) \ 〈a〉Gtn(p)} if this exists. Then there are three posibilities.

(1) |{tn}| = ℵ0. Then

hp(pna) =

{m + n for 0 5 n 5 e1 −m,m + n +

∑ki=1(ci − ti) for ek −m < n 5 ek+1 −m, k = 1,

and

en =

{t1 for n = 1t1 +

∑ni=2(ti − ci−1) for n = 2.

(0.4)

(2) |{ti}| = r for some positive integer r. Then

hp(pna) =

m + n for 0 5 n 5 e1 −m,

m + n +∑k

i=1(ci − ti){

for ek −m < n 5 ek+1 −mand 1 5 k 5 r − 1,

and for n > er −m,

hp(pna) =

{m + n +

∑ri=1(ci − ti) if hp(pna) < ω for all n = 1

∞ if hp(psa) = ω for some integer s = 1

where en is as (0.4).

12 ABSTRACTS

(3) |{ti}| = 0. Thenhp(pna) = m + n

for all n = 0.

Using Corollary0.3 and some previous results, we have the following.

Corollary 0.5. Let G be an abelian group whose torsion part T is torsion-complete.Then all torsion-free rank-one subgroups of G are quasi purifiable in G and hencethe height-matrices of the torsion-free elements of the group G can be computed.

Takashi Okuyama email: [email protected] of MathematicsToba National College of Maritime Technology1-1, Ikegami-cho, Toba-shi, Mie-ken, 517-8501, Japan

Applications of the study of torsion–free modulesto commutative algebra

Bruce Olberding, University of Louisiana at Monroe

ABSTRACT. The techniques and tools from commutative algebra necessarilyinfluence the study of torsion-free modules over commutative integral domains indeep and various ways. The study of torsion–free modules, in turn, has had aninteresting influence on the development of new concepts in commutative algebra.In this talk, we survey some recent work on the study of torsion–free modules thatunifies approaches to two rather different classes of commutative rings: Noetheriandomains and Prufer domains. Emphasis is placed on how examples from theseclasses of Noetherian and Prufer rings arise in natural, and in particular, geometric,settings.

Bruce Olberding email: [email protected] of MathematicsUniversity of Louisiana at MonroeMonroe, LA 71209

Quasi–Baer hulls of rings

Gary F. Birkenmeier, University of Louisiana at LafayetteJae Keol Park, Busan National University

S. Tariq Rizvi, Ohio State University at Lima

ABSTRACT. A ring R with identity is called quasi-Baer if the right annihilatorof every ideal is generated, as a right ideal, by an idempotent. Examples of quasi-Baer rings are: any prime ring, any Baer ring, any semiprime CS-ring, any biregularring whose lattice principal ideals is complete, and any right Martindale ring ofquotients of a semiprime ring. The class of quasi-Baer rings is closed under Moritaequivalence, direct products, triangular matrix ring extensions, various types ofpolynomial extensions, and u.p.-monoid ring extensions. We show that a semiprimering R has a smallest right ring of quotients which is quasi-Baer. We call thisring the quasi-Baer hull of R. The quasi-Baer hull of R is generated by R andthe central idempotents of the right maximal ring of quotients of R. We develop


results and examples of the quasi-Baer hull and consider applications to rings withan involution.

Gary F. BirkenmeierDepartment of MathematicsUniversity of Louisiana at LafayetteLafayette, Louisiana 70504--1010, U. S. A.

Jae Keol Park email: [email protected] of MathematicsBusan National UniversityBusan 609--735, Korea

S. Tariq RizviDepartment of MathematicsOhio State University at LimaLima, Ohio 45804--3576, U. S. A.

Derivations in Banach Modules

Chun-Gil Park, Chungnam National University

ABSTRACT. We prove the stability of the functional equation D(x2) = 2ωxD(x)for Banach module-valued functions on a unital Banach algebra A and a unital C∗-algebra A, where ω is an invertible element in the center of A. This is appliedto show that for an almost generalized derivation from A into a left Banach A-module AM , there exists a unique generalized derivation D(xy) = ωxD(y)+ωyD(x)from A into AM near the almost generalized derivation, and that for an almostgeneralized Poisson bracket from A×A into AM , there exists a unique generalizedPoisson bracket {xy, z} = ωx{y, z}+ωy{x, z} from A×A into AM near the almostgeneralized Poisson bracket.

Chun-Gil Park email: [email protected] of MathematicsChungnam National UniversityTaejon 305-764, South Korea

On the stacked bases theorem and generalizations

K.M. Rangaswamy, University of Colorado, Colorado Springsand Manfred Dugas, Baylor University

ABSTRACT. This talk reviews a generalization of the stacked bases theorem anddescribes a “stacked filtration” theorem for B–2 groups which leads to a definitiveresult for homogeneous completely decomposable groups.

K.M. Rangaswamy email: [email protected] of Mathematics phone: : 719-262-3311University of ColoradoBox 7150, 1420 Austin BluffsColorado Springs, CO 80933-7150

14 ABSTRACTS

Mid–Century in Seattle

Jim Reid, Wesleyan University

James D Reid email: [email protected] of Mathematics phone: 860-685-2174Wesleyan UniversityMiddletown, CT 06459-0128

Pre–abelian clan categories

Fred Richman, Florida Atlantic University

ABSTRACT. Categories of representations of clans without special loops, andwith a linear ordering at each vertex, are studied with an eye toward identifyingthose that have kernels and cokernels. A complete characterization is given forsimple graphs whose vertices have degree at most two.

Fred Richman email: [email protected] of MathematicsFlorida Atlantic UniversityBoca Raton, FL 33431-0991

On the extending properties for fully invariant submodules

S. Tariq Rizvi, The Ohio State University, Lima

ABSTRACT. A module M is said to be (strongly) FI-extending if every fullyinvariant submodule of M is essential in a (fully invariant) direct summand. A ringR is said to be right (strongly) FI-extending if the module RR is (strongly) FI-extending. We examine the relationship of the strongly FI-extending property withthe FI-extending property and other related concepts, in various settings. We inves-tigate the preservation of the strongly FI-extending property with respect to sub-modules, direct summands, direct sums and endomorphism rings. The (strongly)FI-extending property is completely characterized for formal triangular matrix rings.Among consequences, it is shown that the (strongly)FI-extending property is notleft-right symmetric. Other results and examples are provided which help delimitthe theory. (This is joint work with G. F. Birkenmeier and J. K. Park)

S. Tariq Rizvi email: [email protected] of Mathematics phone: 419-995-8211Department of Mathematics fax: 419-995-8094The Ohio State UniversityLima, OH 45804

Abelian Group Theory in Italy

Luigi Salce, Universita di Padova

Luigi Salce email: [email protected]. di Mat. Pura e Appl. phone: +39-049-8275947Via Belzoni 7 fax: +39-049-827589235131 Padova, Italy


The upper central seriesof the maximal normal p-subgroup of Aut(G)

Phill Schultz, University of Western AustraliaMaria Alicia Avino, Univ. Nat. Autonoma di Mexico

ABSTRACT. Let G be a bounded abelian p-group and ∆ the maximal normalp-subgroup of Aut(G). We recently proved that the lower p-central series of ∆could be described in terms of the Loewy sequence of the radical J of End(G). Inthis talk, we solve the harder problem of describing the upper central series of ∆.

This time, we use the annihilating sequence of J to construct an ascendingsequence in ∆. We say G is tidy if this sequence is the upper central series. Itturns out that G is tidy unless G is a very special kind of 2-group.

Associate Professor Phill Schultz email: [email protected] Honorary Research Fellow phone: (08)9380-3381Dept. of Math. and Stat. fax: (08)9380-1028The University of Western Australia http://www.maths.uwa.edu.au/~schultz/35 Stirling Highway, Nedlands, 6009, Australia

A problem of D. Arnold andsubprojective representations of posets over uniserial algebras

Daniel Simson, Nicholas Copernicus University, Torun, Poland

ABSTRACT. Part 1. Filtered chain categories. Let R be either the ring Z/pmZ,or the uniserial K-algebra Fm = K[t]/(tm), where p ≥ 2 a prime, m ≥ 1 is aninteger and K is a field. Following D. Arnold [1] and [2], given an integer s ≥ 1we consider the filtered chain category C(s, R) whose objects are filtered s-chainsC = (C1 ⊆ C2 ⊆ . . . ⊆ Cs−1 ⊆ Cs) of finitely generated R-modules C1, . . . , Cs, anda morphism from C to C′ in C(s, R) is an R-module homomorphism f : Cs → C′

s

such that f(Cj) ⊆ C′j for j = 1, . . . , s− 1.

One can show that C(s, R) is an additive Krull-Schmidt category with enoughrelative projective objects and enough relative injective objects, and that C(s, R)has almost split sequences. Following [4, Corollary 5.7], one shows that for any Ras above, relative projective objects in C(s, R) are relative injective, and relativeinjective objects in C(s, R) are relative projective. In [2], D. Arnold is interestedin the problem, when the category C(s, R) is of finite, tame or wild representationtype, where the tame type and the wild type are well-defined at least in caseR = K[t]/(tm) and K is algebraically closed.

In the case R = Fm = K[t]/(tm), K is algebraically closed and m ≥ 1, we givea complete solution of the problem in the following three theorems.

Theorem 1.1. The category C(s, Fm) is of finite representation type if and onlyif the pair (m, s) of integers satisfies any of the following conditions:

(F-1) m ≥ 1 or s = 1,(F-2) m ≤ 5 and s = 2,(F-3) m ≤ 3 and 3 ≤ s ≤ 4,

16 ABSTRACTS

(F-4) m = 2 and s ≥ 5. �

Theorem 1.2. The category C(s, Fm) is of wild representation type if and onlyif the pair (m, s) of integers satisfies any of the following conditions:

(W-1) m ≥ 7 and s ≥ 2,(W-2) m ≥ 5 and s ≥ 3,(W-3) m ≥ 4 and s ≥ 5,(W-4) m ≥ 3 and s ≥ 6. �

Theorem 1.3. Assume that the category C(s, Fm) is of infinite representationtype. Then the following three conditions are equivalent:

(a) The category C(s, Fm) is of tame representation type.(b) The category C(s, Fm) is tame of polynomial growth.(c) The pair (m, s) of integers satisfies any of the following conditions:

(T-1) m = 6 and s = 2,(T-2) m = 4 and 3 ≤ s ≤ 4,(T-3) m = 3 and s = 5. �

Our Theorems 1.1, 1.2 and 1.3 are deduced from corresponding results on sub-projective representations of finite posets defined below (see [5], [6] for details).

Part 2. Subprojective representations of posets. Assume that I is a finite posetwith a unique maximal element ?. Let F be a commutative ring. In [5] and [6]we have defined a filtered subprojective F -representation of I to be a systemX = (Xj)j∈I of finitely generated F -modules satisfying the following conditions:

(a) X? is a projective F -module,(b) Xj is a submodule of X? for every j ∈ I, and Xi ⊆ Xj , if i � j in I.By a morphism f : X → X ′ of filtered subprojective F -representations X and

X ′ of the poset I we mean an F -module homomorphism f : X? → X ′? such that

f(Xj) ⊆ X ′j for every j ∈ I.

We denote by fspr(I, F ) the category of filtered subprojective F -representationsof the poset I over F .

On shows that fspr(I, F ) is an additive Krull-Schmidt category with enough rel-ative projective objects and enough relative injective objects. If F is artinian thenfspr(I, F ) has Auslander-Reiten sequences. If F is a finite dimensional algebra overan algebraically closed field K then one defines the tame representation type, thepolynomial growth and the wild representation type for fspr(I, F ) as for the cate-gory mod(A) of finite dimensional A-modules over a finite dimensional K-algebraA. Moreover, one proves that any such a category fspr(I, F ) is either tame or wild.

Let m ≥ 1 and let Fm = K[t]/(tm), where K is an algebraically closed field.We shall present complete lists of pairs (I, m), where I is a poset as above andm ≥ 1, for which the category fspr(I, Fm) is of tame representation type (resp.of finite representation type, of wild representation type, or fspr(I, Fm) is tame ofnon-polynomial growth).

Theorem 2.1. Assume that m ≥ 2 and |I| ≥ 2. The category fspr(I, Fm) is offinite representation type if and only if the pair (m, I) satisfies any of the followingconditions:

1◦ I is linearly ordered and the pair (m, |I| − 1) satisfies any of the conditions(F-1)-(F-4) of Theorem 1.1, or


2◦ I is not linearly ordered and the pair (m, |I|−1) satisfies any of the conditions(F-1)-(F-4) of Theorem 1.1, or

(F-5) m = 3 and I = (• → ?← •), or(F-6) m = 2 and I is a subposet of any of the posets F1, . . . ,F7 presented in

[6]. �

Theorem 2.2. Assume that m ≥ 2 and |I| ≥ 2. The category fspr(I, Fm) is oftame representation type if and only if the pair (m, I) of any of the following types:

0◦ (m, |I|) is in Theorem 2.1, or1◦ I is linearly ordered and the pair (m, |I| − 1) satisfies any of the conditions

(T-1), (T-2), (T-3) of Theorem 1.2, or2◦ I is not linearly ordered, I does not contain four incomparable points and the

pair (m, I) satisfies any of the following conditions:(T-4) m = 4 and I = (• → ?← •), or(T-5) m = 3 and I is any of the posets

•↗ ↘

• → • → ? ,

•↘

• → • → ?, or

(T-6) m = 2, I is not of type (F-6) of Theorem 2.1, and I is a one-peaksubposet of any of the posets T1, . . . , T21 presented in [6, Table 1]. �

The general idea of the proof of Theorems 2.1 and 2.2 is outlined in [5, Section5], where in fact Theorem 1.1 is proved. One of the main reduction tools we applyis a covering type functor Im-spK → fspr(I, Fm) described in [5] and [6]. It relatesthe category fspr(I, Fm) with the category Im-spK of Im-spaces over the field K,where Im is an infinite poset associated with (I, m) in [5, Section 5].

Note that Theorem 2.1 can be also deduced from the main result of Plohotnik[3].

References

[1] D. M. Arnold, Representations of partially ordered sets and abelian groups, ContemporaryMath. 87(1989), 91-109.

[2] D. M. Arnold, ”Abelian Groups and Representations of Finite Partially Ordered Sets”,Canad. Math. Soc. Books in Math., Springer-Verlag, New York Berlin Heidelberg, 2000.

[3] V. V. Plahotnik, Representations of partially ordered sets over commutative rings, Izv.Akad. Nauk SSSR 40 (1976), 527 - 543 (in Russian).

[4] D. Simson, ”Linear Representations of Partially Ordered Sets and Vector Space Cate-gories”, Algebra, Logic and Applications, Vol. 4, Gordon & Breach Science Publishers,1992.

[5] D. Simson, Socle projective representations of partially ordered sets and Tits quadraticforms with application to lattices over orders, in Proceedings of the Conference on AbelianGroups and Modules, Colorado Springs, August 1995, Lecture Notes in Pure and Appl.Math., Vol. 182, 1996, pp. 73-111.

[6] D. Simson, A covering functor and tame representation type for poset representations overartinian principal ideal algebras, N. Copernicus University, Preprint, 1996.

Daniel Simson email: [email protected] Copernicus UniversityFaculty of Mathematics and Computer Scienceul. Chopina 12/1887--100 Torun, Poland

18 ABSTRACTS

A consistency result on infinite rank Butler groups

Lutz Strungmann, University of Essen and The Hebrew University

ABSTRACT. Both, B1-groups and B2-groups are natural generalizations offinite rank Butler groups to the infinite rank case and it is known that every B2-group is a B1-group. Moreover, assuming V = L it was proven that the two classescoincide. Here we demonstrate that it is undecidable in ZFC whether or not allB1-groups are B2-groups. Using Cohen forcing we prove that there is a model ofZFC in which there exists a B1-group that is not a B2-group.

Lutz Struengmann email: [email protected] for Mathematics departmental fax: +972(0)2 563 0702The Hebrew University home phone: +972(0)2 627 3665Givat Ram office phone: +972(0)2 658 6816Jerusalem 91904, Israel

Noetherian Generalised Power Series Rings and Modules

K. Varadarajan, University of Calgary

K. Varadarajan email: [email protected] of Mathematics and StatisticsUniversity of CalgaryCalgary, AlbertaCANADA. T2N 1N4.

K0–like constructions for almost completely decomposable groups

Charles Vinsonhaler, University of Connecticut

ABSTRACT. In an earlier paper the authors introduced a K0-like constructionthat produces, for each torsion-free abelian group A of finite rank, a finitely gen-erated abelian group G(A). In this note, we show that for any finite abelian groupS, there is an almost completely decomposable (acd) group A such that G(A) hastorsion subgroup isomorphic to S. In addition, if S is a finitely generated abeliangroup satisfying a certain condition on the torsion-free rank, then there is an al-most completely decomposable group A such that G(A) ' S. In the usual K0

construction for acd groups, one always obtains a trivial torsion subgroup. Thus,G(A) appears to be a more versatile tool than K0 for the study of acd’s.

Charles I. Vinsonhaler email: [email protected] of Mathematics office phone: 860-486-3944University of Connecticut196 Auditorium Road U-9Storrs, CT 06269-3009


Abelian Groups at New Mexico State University

Elbert Walker, New Mexico State University

ABSTRACT. This talk is an informal description of how the mathematicsdepartment at New Mexico State University came to have an emphasis in Abeliangroup theory. The story starts in the late 1950’s. Mention will be made of themany long term and short term faculty, visitors, and students. Some discussionof topics studied will be made, but mainly this talk is non-technical, more aboutpeople and events than theorems.

Elbert A Walker email: [email protected] of Mathematics phone: 505-646-2707New Mexico State UniversityLas Cruces, NM 88003-0105

Multi–isomorphism for mixed groups

William J. Wickless, University of Connecticut

ABSTRACT. Groups A, B are multi-isomorphic if An ∼= Bn for all n ≥ 2. Ina series of papers, O’Meara and Vinsonhaler have studied this relation in the classT F of all torsion-free finite rank groups. I study this notion in the class D of allquotient divisible (qd) groups. A reduced group A is qd if there exists a finite rankfree subgroup F ≤ A such that A/F is a divisible torsion group.

If we are willing to replace qd groups by their equivalence classes under a mildequivalence relation, the following theorem, proved in the tffr case by O’Meara andVinsonhaler, holds: qd groups A, B are multi-isomorphic if and only if A ⊕ A ∼=A ⊕ B. However, the classes D and T F are far from identical, even though thereis a duality from D and quasi-homomorphisms to T F and quasi-homomorphisms.Thus, it comes as no surprise that, contrary to the torsion-free case, there is a qdgroup A such that A ⊕ A ∼= A ⊕ B implies A ∼= B but QE(A) is isomorphic to aquaternion algebra.

For a qd group A, let P (A) be the class of groups B such that B is A-projectiveand A is B-projective. As in the torsion-free case, P (A) is a cancellative semigroupunder direct sum. Then one can introduce a K0 type group G(A). We computeG(A) in a number of cases. It seems that the computation of G(A) is easier in theqd case.

William J. Wickless email: [email protected] of MathematicsUniversity of Connecticut196 Auditorium Road, U-9Storrs, CT 06269-3009

20 ABSTRACTS

Certain additive subgroups on prime rings

Tsai-Lien Wong, National Sun Yat-sen University

Tsai-Lien Wong email: [email protected]. of Applied MathematicsNational Sun Yat-sen UniversityKaohsiung, Taiwan 00804

On Three Open Questions on Quasi–Frobenius Rings

Mohamed F. Yousif, The Ohio State University

Mohamed F. Yousif email: [email protected] Ohio State University phone: 1-419-995-8368Lima, Ohio 45804 fax: 1-419-995-8094

u–indepencence and quadratic u-independence in the constructionof indecomposable finitely generated modules

Paolo Zanardo, Universita di Padova

ABSTRACT. Let R be a valuation domain, and let S be a fixed maximal imme-diate extension of R. There is a somewhat standard way to define finitely generatedR-modules M by generators and relations, relating M with a set of units u1, . . . , un

of S. The notion of u-independence of units u1, . . . , un of S over an ideal I of R wasintroduced by Zanardo in 1985. It allowed to show the existence of indecomposablefinitely generated R-modules M (related to u1, . . . , un) with minimal number ofgenerators l(M) = n + 1 and Goldie dimension g(M) = n. This solved the prob-lem of finding indecomposable finitely generated R-modules with Goldie dimensiongreater than one. However, it is worth noting that the argument developed in thatpaper worked exactly in the case when l(M) = g(M) + 1.

M. D. Lunsford in 1995 gave a natural extension of u-independence definingquadratic u-independence of units u1, . . . , un of S over an ideal I. Starting witha convenient set of units of S, for any pair of positive integers h, k he definedby generators and relations an R-module M with l(M) = h + k and g(M) =h. Using quadratic u-independence Lunsford was able to prove that such M isindecomposable: in fact, he proved much more, namely that EndR(M) is a localring.

The notion of quadratic u-independence over an ideal I has a slight inconve-nience: it works only in case I is a prime ideal of R (necessarily non-zero andnon-maximal, in this setting). And in general, even if I is a prime ideal, youmay have large sets of u-independent elements, and no quadratic u-independence.Therefore, Lunsford’s results cannot prove the existence of indecomposable finitelygenerated modules M with 1 < g(M) < l(M) − 1 for large classes of valuationdomains R (for instance, archimedean valuation domains, where the only primesare the maximal ideal and zero).

We show that the existence of a set of 4 units of S which are u-independentover any nonzero ideal I is enough to ensure the existence of an indecomposableR-module M with l(M) = 4 and g(M) = 2. The definition of M by generatorsand relations is the same as in Lunsford’s paper. But, since the related units of S


are just u-independent and not quadratically u-independent, we need a completelydifferent, more direct argument to prove that M is indecomposable.

Paolo Zanardo email: [email protected]. di Mat. Pura e Appl. phone: +39-049-8275947Via Belzoni 7 fax: +39-049-827589235131 Padova, Italy

An annihilator condition of modules

Yiqiang Zhou, Memorial University of Newfoundland

Yiqiang Zhou email: [email protected] of Mathematics and StatisticsMemorial University of NewfoundlandSt.John’s, A1C 5S7, Canada

Documents

SECOND HONOLULU CONFERENCE - University of Hawaiiadolf/Meeting/abstracts.pdf · 2001-09-24 · SECOND HONOLULU CONFERENCE ABSTRACTS ... In this paper we show an extended version of