1st Edition

Abelian Groups, Rings, Modules, and Homological Algebra

Edited By Pat Goeters, Overtoun M.G. Jenda Copyright 2006
    360 Pages 15 B/W Illustrations
    by Chapman & Hall

    360 Pages
    by Chapman & Hall

    About the book…

    In honor of Edgar Enochs and his venerable contributions to a broad range of topics in Algebra, top researchers from around the world gathered at Auburn University to report on their latest work and exchange ideas on some of today's foremost research topics. This carefully edited volume presents the refereed papers of the participants of these talks along with contributions from other veteran researchers who were unable to attend.

    These papers reflect many of the current topics in Abelian Groups, Commutative Algebra, Commutative Rings, Group Theory, Homological Algebra, Lie Algebras, and Module Theory. Accessible even to beginning mathematicians, many of these articles suggest problems and programs for future study. This volume is an outstanding addition to the literature and a valuable handbook for beginning as well as seasoned researchers in Algebra.

    about the editors…

    H. PAT GOETERS completed his undergraduate studies in mathematics and computer science at Southern Connecticut State University and received his Ph.D. in 1984 from the University of Connecticut under the supervision of William J. Wickless. After spending one year in a post-doctoral position in Wesleyan University under the tutelage of James D. Reid, Goeters was invited for a tenure track position in Auburn University by Ulrich F. Albrecht. Soon afterwards, William Ullery and Overtoun Jenda were hired, and so began a lively Algebra group.


    OVERTOUN M. G. JENDA received his bachelor's degree in Mathematics from Chancellor College, the University of Malawi. He moved to the U.S. 1977 to pursue graduate studies at University of Kentucky, earning his Ph.D. in 1981 under the supervision of Professor Edgar Enochs. He then returned to Chancellor College, where he was a lecturer (assistant professor) for three years. He moved to the University of Botswana for another three-year stint as a lecturer before moving back to the University of Kentucky as a visiting assistant professor in 1987. In 1988, he joined the Algebra research group at Auburn University.

    GENERALIZING WARFIELD'S HOM AND TENSOR RELATIONS, Ulrich Albrecht and Pat Goeters
    Introduction1
    Self-Small Modules
    Projectivity Properties
    The Class MA
    Domains Which Support Warfield's Results
    Replicating Duality for Domains
    Duality and Infinite Products
    Mixed Groups

    HOW FAR IS AN HFD FROM A UFD?, David F Anderson and Elizabeth V Mclaughlin
    Introduction
    3R/
    Localization
    Questions

    A COUNTER EXAMPLE FOR A QUESTION ON PSEUDO-VALUATION RINGS, Ayman Badawi
    Introduction
    Counter example

    CO-LOCAL SUBGROUPS OF ABELIAN GROUPS, Joshua Buckner and Manfred Dugas
    Introduction
    Basic Properties
    Cotorsion-free groups as co-local subgroups

    PARTITION BASES AND B1- GROUPS, Immacolata Caruso, Clorinda De Vivo and ClaudiaMetelli
    Introduction
    Preliminaries
    Partition bases
    Direct Summands
    The Domain of C;D
    Indecomposable summands
    Examples

    ASSOCIATED PRIMES OF THE LOCAL COHOMOLOGY MODULES, Mohammad T Dibaei and Siamak Yassemi
    Introduction
    General case
    Special case
    Generalized local cohomology

    ON INVERSE LIMITS OF B´EZOUT DOMAINS, David E Dobbs and Marco Fontana
    Introduction
    Results

    AN ELEMENTARY PROOF OF GROTHENDIECK'S THEOREM, E Enochs, S Estrada and B Torrecillas
    Introduction
    The main theorem
    Grothendieck's Theorem

    GORENSTEIN HOMOLOGICAL ALGEBRA, Edgar E Enochs and Overtoun MG Jenda
    Introduction
    Tate Homology and Cohomology
    Auslander and Gorenstein Rings
    The Kaplansky Program
    Iwanaga-Gorenstein Rings
    Gorenstein Homological Algebra
    Generalized Tate Homology and Cohomology
    The Avramov-Martsinkovsky Program
    Gorenstein Flat Modules
    Salce's Cotorsion Theories
    Other Possibilities

    MODULES AND POINT SET TOPOLOGICAL SPACES, Theodore G Faticoni
    The Diagram
    Self-small and Self-slender Modules
    The Construction Function
    The Greek Maps
    Coherent Modules and Complexes
    Complete Setsof Invariants
    Unique Decompositions
    Homological Dimensions
    Miscellaneous

    INJECTIVEMODULES AND PRIME IDEALS OF UNIVERSAL ENVELOPING ALGEBRAS, J¨org Feldvoss
    Injective Modules and Prime Ideals
    InjectiveHulls
    Locally Finite Submodules of the Coregular Module
    Minimal Injective Resolutions

    COMMUTATIVE IDEAL THEORY WITHOUT FINITENESS CONDITIONS, Laszlo Fuchs, William Heinzer and Bruce Olberding
    Introduction
    The structure of Q-irreducible ideals
    Completely Q-irreducible and m-canonical ideals
    Q-irreducibility and injective modules
    Irredundant decompositions and semi-artinian modules
    Pr¨uferdomains
    Questions
    Appendix:Corrections to17

    COVERS AND RELATIVE PURITY OVER COMMUTATIVE NOETHERIAN LOCAL RINGS, JR Garc´ia Rozas, L Oyonarte and B Torrecillas
    Preliminaries
    tI -closed modules
    Relative purity over local rings
    Relative purity over regular local rings

    TORSIONLESS LINEARLY COMPACT MODULES, R¨udiger G¨obel and Saharon Shelah
    Introduction
    Proof of the Theorem

    BIG INDECOMPOSABLE MIXED MODULES OVER HYPERSURFACE SINGULARITIES, Wolfgang Hassler and Roger Wiegand
    Introduction
    Bimodules
    Extensions
    Syzygies and double branched covers
    Finding a suitable finite-length module
    The main application

    EVERY ENDOMORPHISM OF A LOCAL WARFIELD MODULE IS THE SUM OF TWO AUTOMORPHISMS, Paul Hill, Charles Megibben and William Ullery
    Introduction
    The Key Lemma
    Proof of the Main Theorem

    WAKAMATSU TILTING MODULES, U-DOMINANT DIMENSION AND K-GORENSTEIN MODULES, Zhaoyong Huang
    Introduction and main results
    Wakamatsu tilting modules
    The proof of main results
    Exactness of the double dual
    A generalization of k-Gorenstein modules

    G-SEPARATED COVERS, Lawrence S Levy and Jan Trlifaj
    Introduction
    G-covers
    G-separated covers
    The Dedekind-likecase

    OpenProblems
    THE COTORSION DIMENSION OF MODULES AND RINGS, Lixin Mao and Nanqing Ding
    Introduction
    General results
    Cotorsion dimension under change of rings
    Applications incommutative rings

    MAXIMAL SUBRINGS OF HOMOGENEOUS FUNCTIONS, C J Maxson
    Introduction
    The Case of Torsion Groups
    The Case of Torsion-Free Groups
    Subrings of M0(A)

    ISOTYPE SEPARABLE SUBGROUPS OF MIXED ABELIAN GROUPS, Charles Megibben and William Ullery
    Introduction
    Subgroups with _-covers of almost balanced pure subgroups
    Intersection closure of global Warfield groups
    Isotype separable subgroups of globalWarfield groups

    NOTE ON THE GENERALIZED DERIVATION TOWER THEOREM FOR LIE ALGEBRAS, Toukaiddine Petit and Fred Van Oystaeyen
    Introduction
    G-Decomposition
    Derivation tower of Lie algebras: case with trivialc enter
    The Derivation tower of Lie algebras: general case

    QUOTIENT DIVISIBLE GROUPS, !-GROUPS, AND AN EXAMPLE OF FUCHS, J D Reid
    Introduction
    On w-groups
    Three Remarks
    Parameters
    Main Results
    Endomorphisms

    WHEN ARE ALMOST PERFECT DOMAINS NOETHERIAN?, Luigi Salce
    Introduction
    Known results on the Noetherian condition
    A characterization of Noetherian almost perfect domains
    E-closed domains

    PURE INVARIANCE IN TORSION-FREE ABELIAN GROUPS, Phill Schultz
    Introduction
    Pure fully invariant subgroups
    Traces and kernels of cd groups

    COMPRESSIBLE AND RELATED MODULES, Patrick F Smith
    Introduction
    Prime and compressible modules
    Monoform modules
    Nonsingular modules
    Fully bounded rings

    Biography

    Pat Goeters, Overtoun M.G. Jenda