1st Edition

# Abelian Groups, Rings, Modules, and Homological Algebra

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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.

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