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Abelian Groups, Rings, Modules, and Homological Algebra





ISBN 9781584885528
Published January 27, 2006 by Chapman and Hall/CRC
360 Pages 15 B/W Illustrations

 
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Book Description

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.

Table of Contents

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

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