Arithmetical Properties of Commutative Rings and Monoids 1st Edition Scott T. Chapman Arithmetical Properties of Commutative Rings and Monoids 1st Edition Scott T. Chapman Arithmetical Properties of Commutative Rings and Monoids 1st Edition Scott T. Chapman

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7. Arithmetical Properties
of Commutative Rings
and Monoids

8. M. S. Baouendi
University of California,
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Jane Cronin
Rutgers University
Jack K. Hale
Georgia Institute of Technology
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University of California,
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Yale University
Anil Nerode
Cornell University
Donald Passman
University of Wisconsin,
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4. F. Hirzebruch et al., Differentiable Manifolds and Quadratic Forms
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7. B. R. McDonald et al., Ring Theory
8. Y.-T. Siu, Techniques of Extension on Analytic Objects
9. S. R. Caradus et al., Calkin Algebras and Algebras of Operators
on Banach Spaces
10. E. O. Roxin et al., Differential Games and Control Theory
11. M. Orzech and C. Small, The Brauer Group of Commutative Rings
12. S. Thomier, Topology and Its Applications
13. J. M. Lopez and K. A. Ross, Sidon Sets
14. W. W. Comfort and S. Negrepontis, Continuous Pseudometrics
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27. R. B. Mura and A. Rhemtulla, Orderable Groups
28. J. R. Graef, Stability of Dynamical Systems
29. H.-C. Wang, Homogeneous Branch Algebras
30. E. O. Roxin et al., Differential Games and Control Theory II
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32. M. Altman, Contractors and Contractor Directions Theory and Applications
33. J. S. Golan, Decomposition and Dimension in Module Categories
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35. J. D. Sally, Numbers of Generators of Ideals in Local Rings
36. S. S. Miller, Complex Analysis
37. R. Gordon, Representation Theory of Algebras
38. M. Goto and F. D. Grosshans, Semisimple Lie Algebras
39. A. I. Arruda et al., Mathematical Logic
40. F. Van Oystaeyen, Ring Theory
41. F. Van Oystaeyen and A. Verschoren, Reflectors and Localization
42. M. Satyanarayana, Positively Ordered Semigroups
43. D. L Russell, Mathematics of Finite-Dimensional Control Systems
44. P.-T. Liu and E. Roxin, Differential Games and Control Theory III
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47. P.-T. Liu and J. G. Sutinen, Control Theory in Mathematical Economics
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54. R. L. Sternberg et al., Nonlinear Partial Differential Equations
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55. B. R. McDonald, Ring Theory and Algebra III
56. J. S. Golan, Structure Sheaves Over a Noncommutative Ring
57. T. V. Narayana et al., Combinatorics, Representation Theory
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58. T. A. Burton, Modeling and Differential Equations in Biology
59. K. H. Kim and F. W. Roush, Introduction to Mathematical
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60. J. Banas and K. Goebel, Measures of Noncompactness in Banach Spaces
61. O. A. Nielson, Direct Integral Theory
62. J. E. Smith et al., Ordered Groups
63. J. Cronin, Mathematics of Cell Electrophysiology
64. J. W. Brewer, Power Series Over Commutative Rings
65. P. K. Kamthan and M. Gupta, Sequence Spaces and Series
66. T. G. McLaughlin, Regressive Sets and the Theory of Isols
67. T. L. Herdman et al., Integral and Functional Differential Equations
68. R. Draper, Commutative Algebra
69. W. G. McKay and J. Patera, Tables of Dimensions, Indices,
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70. R. L. Devaney and Z. H. Nitecki, Classical Mechanics
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71. J. Van Geel, Places and Valuations in Noncommutative Ring Theory
72. C. Faith, Injective Modules and Injective Quotient Rings
73. A. Fiacco, Mathematical Programming with Data Perturbations I
74. P. Schultz et al., Algebraic Structures and Applications
75. L Bican et al., Rings, Modules, and Preradicals
76. D. C. Kay and M. Breen, Convexity and Related Combinatorial Geometry
77. P. Fletcher and W. F. Lindgren, Quasi-Uniform Spaces
78. C.-C. Yang, Factorization Theory of Meromorphic Functions
79. O. Taussky, Ternary Quadratic Forms and Norms
80. S. P. Singh and J. H. Burry, Nonlinear Analysis and Applications
81. K. B. Hannsgen et al., Volterra and Functional Differential Equations
82. N. L. Johnson et al., Finite Geometries
83. G. I. Zapata, Functional Analysis, Holomorphy, and Approximation Theory
84. S. Greco and G. Valla, Commutative Algebra
85. A. V. Fiacco, Mathematical Programming with Data Perturbations II
86. J.-B. Hiriart-Urruty et al., Optimization
87. A. Figa Talamanca and M. A. Picardello, Harmonic Analysis on Free Groups
88. M. Harada, Factor Categories with Applications to Direct Decomposition
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89. V. I. Istra’tescu, Strict Convexity and Complex Strict Convexity
90. V. Lakshmikantham, Trends in Theory and Practice of Nonlinear Differential
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93. J. W. Longley, Least Squares Computations Using Orthogonalization
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94. L. P. de Alcantara, Mathematical Logic and Formal Systems
95. C. E. Aull, Rings of Continuous Functions
96. R. Chuaqui, Analysis, Geometry, and Probability
97. L. Fuchs and L. Salce, Modules Over Valuation Domains
98. P. Fischer and W. R. Smith, Chaos, Fractals, and Dynamics
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101. R.-E. Hoffmann and K. H. Hofmann, Continuous Lattices
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102. J. H. Lightbourne III and S. M. Rankin III, Physical Mathematics
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104. J. W. Brewer et al., Linear Systems Over Commutative Rings
105. C. McCrory and T. Shifrin, Geometry and Topology
106. D. W. Kueke et al., Mathematical Logic and Theoretical Computer Science
107. B.-L. Lin and S. Simons, Nonlinear and Convex Analysis
108. S. J. Lee, Operator Methods for Optimal Control Problems
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112. D. V. Chudnovsky and G. V. Chudnovsky, Search Theory
113. D. V. Chudnovsky and R. D. Jenks, Computer Algebra
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115. P. Nelson et al., Transport Theory, Invariant Imbedding,
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116. P. Clément et al., Semigroup Theory and Applications
117. J. Vinuesa, Orthogonal Polynomials and Their Applications
118. C. M. Dafermos et al., Differential Equations
119. E. O. Roxin, Modern Optimal Control
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131. O. Arino et al., Mathematical Population Dynamics
132. S. Coen, Geometry and Complex Variables
133. J. A. Goldstein et al., Differential Equations with Applications in Biology,
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135. P Clément et al., Semigroup Theory and Evolution Equations

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137. J. M. Bayod et al., p-adic Functional Analysis
138. G. A. Anastassiou, Approximation Theory
139. R. S. Rees, Graphs, Matrices, and Designs
140. G. Abrams et al., Methods in Module Theory
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145. T. Mabuchi and S. Mukai, Einstein Metrics and Yang–Mills Connections
146. L. Fuchs and R. Göbel, Abelian Groups
147. A. D. Pollington and W. Moran, Number Theory with an Emphasis
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148. G. Dore et al., Differential Equations in Banach Spaces
149. T. West, Continuum Theory and Dynamical Systems
150. K. D. Bierstedt et al., Functional Analysis
151. K. G. Fischer et al., Computational Algebra
152. K. D. Elworthy et al., Differential Equations, Dynamical Systems,
and Control Science
153. P.-J. Cahen, et al., Commutative Ring Theory
154. S. C. Cooper and W. J. Thron, Continued Fractions
and Orthogonal Functions
155. P. Clément and G. Lumer, Evolution Equations, Control Theory,
and Biomathematics
156. M. Gyllenberg and L. Persson, Analysis, Algebra, and Computers
in Mathematical Research
157. W. O. Bray et al., Fourier Analysis
158. J. Bergen and S. Montgomery, Advances in Hopf Algebras
159. A. R. Magid, Rings, Extensions, and Cohomology
160. N. H. Pavel, Optimal Control of Differential Equations
161. M. Ikawa, Spectral and Scattering Theory
162. X. Liu and D. Siegel, Comparison Methods and Stability Theory
163. J.-P. Zolésio, Boundary Control and Variation
164. M. Kr’íz’’ek et al., Finite Element Methods
165. G. Da Prato and L. Tubaro, Control of Partial Differential Equations
166. E. Ballico, Projective Geometry with Applications
167. M. Costabel et al., Boundary Value Problems and Integral Equations
in Nonsmooth Domains
168. G. Ferreyra, G. R. Goldstein, and F. Neubrander, Evolution Equations
169. S. Huggett, Twistor Theory
170. H. Cook et al., Continua
171. D. F. Anderson and D. E. Dobbs, Zero-Dimensional Commutative Rings
172. K. Jarosz, Function Spaces
173. V. Ancona et al., Complex Analysis and Geometry
174. E. Casas, Control of Partial Differential Equations and Applications
175. N. Kalton et al., Interaction Between Functional Analysis, Harmonic
Analysis, and Probability
176. Z. Deng et al., Differential Equations and Control Theory
177. P. Marcellini et al. Partial Differential Equations and Applications
178. A. Kartsatos, Theory and Applications of Nonlinear Operators of Accretive
and Monotone Type

13. 179. M. Maruyama, Moduli of Vector Bundles
180. A. Ursini and P. Aglianò, Logic and Algebra
181. X. H. Cao et al., Rings, Groups, and Algebras
182. D. Arnold and R. M. Rangaswamy, Abelian Groups and Modules
183. S. R. Chakravarthy and A. S. Alfa, Matrix-Analytic Methods
in Stochastic Models
184. J. E. Andersen et al., Geometry and Physics
185. P.-J. Cahen et al., Commutative Ring Theory
186. J. A. Goldstein et al., Stochastic Processes and Functional Analysis
187. A. Sorbi, Complexity, Logic, and Recursion Theory
188. G. Da Prato and J.-P. Zolésio, Partial Differential Equation Methods
in Control and Shape Analysis
189. D. D. Anderson, Factorization in Integral Domains
190. N. L. Johnson, Mostly Finite Geometries
191. D. Hinton and P. W. Schaefer, Spectral Theory and Computational
Methods of Sturm–Liouville Problems
192. W. H. Schikhof et al., p-adic Functional Analysis
193. S. Sertöz, Algebraic Geometry
194. G. Caristi and E. Mitidieri, Reaction Diffusion Systems
195. A. V. Fiacco, Mathematical Programming with Data Perturbations
196. M. Kr’íz’cek et al., Finite Element Methods: Superconvergence,
Post-Processing, and A Posteriori Estimates
197. S. Caenepeel and A. Verschoren, Rings, Hopf Algebras, and Brauer Groups
198. V. Drensky et al., Methods in Ring Theory
199. W. B. Jones and A. Sri Ranga, Orthogonal Functions, Moment Theory,
and Continued Fractions
200. P. E. Newstead, Algebraic Geometry
201. D. Dikranjan and L. Salce, Abelian Groups, Module Theory, and Topology
202. Z. Chen et al., Advances in Computational Mathematics
203. X. Caicedo and C. H. Montenegro, Models, Algebras, and Proofs
204. C. Y. Yédérém and S. A. Stepanov, Number Theory and Its Applications
205. D. E. Dobbs et al., Advances in Commutative Ring Theory
206. F. Van Oystaeyen, Commutative Algebra and Algebraic Geometry
207. J. Kakol et al., p-adic Functional Analysis
208. M. Boulagouaz and J.-P. Tignol, Algebra and Number Theory
209. S. Caenepeel and F. Van Oystaeyen, Hopf Algebras and Quantum Groups
210. F. Van Oystaeyen and M. Saorin, Interactions Between Ring Theory
and Representations of Algebras
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212. T.-X. He, Wavelet Analysis and Multiresolution Methods
213. H. Hudzik and L. Skrzypczak, Function Spaces: The Fifth Conference
214. J. Kajiwara et al., Finite or Infinite Dimensional Complex Analysis
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in Physical and Life Sciences
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217. J. Herzog and G. Restuccia, Geometric and Combinatorial Aspects
of Commutative Algebra
218. G. Chen et al., Control of Nonlinear Distributed Parameter Systems
219. F. Ali Mehmeti et al., Partial Differential Equations on Multistructures
220. D. D. Anderson and I. J. Papick, Ideal Theoretic Methods
in Commutative Algebra
221. Á. Granja et al., Ring Theory and Algebraic Geometry
222. A. K. Katsaras et al., p-adic Functional Analysis

14. 223. R. Salvi, The Navier-Stokes Equations
224. F. U. Coelho and H. A. Merklen, Representations of Algebras
225. S. Aizicovici and N. H. Pavel, Differential Equations and Control Theory
226. G. Lyubeznik, Local Cohomology and Its Applications
227. G. Da Prato and L. Tubaro, Stochastic Partial Differential Equations
and Applications
228. W. A. Carnielli et al., Paraconsistency
229. A. Benkirane and A. Touzani, Partial Differential Equations
230. A. Illanes et al., Continuum Theory
231. M. Fontana et al., Commutative Ring Theory and Applications
232. D. Mond and M. J. Saia, Real and Complex Singularities
233. V. Ancona and J. Vaillant, Hyperbolic Differential Operators
and Related Problems
234. G. R. Goldstein et al., Evolution Equations
235. A. Giambruno et al., Polynomial Identities and Combinatorial Methods
236. A. Facchini et al., Rings, Modules, Algebras, and Abelian Groups
237. J. Bergen et al., Hopf Algebras
238. A. C. Krinik and R. J. Swift, Stochastic Processes and Functional Analysis:
A Volume of Recent Advances in Honor of M. M. Rao
239. S. Caenepeel and F. van Oystaeyen, Hopf Algebras in Noncommutative
Geometry and Physics
240. J. Cagnol and J.-P. Zolésio, Control and Boundary Analysis
241. S. T. Chapman, Arithmetical Properties of Commutative Rings
and Monoids

15. Boca Raton London New York Singapore
Arithmetical
Properties
of Commutative
Rings and Monoids
Scott T. Chapman
Trinity University
San Antonio, Texas, U.S.A.

16. Published in 2005 by
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17. Preface
This volume contains the Proceedings of two related meetings which took place
in October 2003. The first event was a one day Mini-Conference on Factorization
Properties of Commutative Rings and Monoids hosted by the Department of Math-
ematics at the University of North Carolina at Chapel Hill. The Mini-Conference
was followed by a regular Special Session on Commutative Rings and Monoids at
the Fall 2003 Southeast Sectional Meeting of the American Mathematical Society.
A major focus of these two meetings was the study of properties related to
nonunique factorizations of elements into irreducible elements in commutative rings
and monoids. Such problems have emerged as an independent area of research only
over the past 30 years. Almost every result obtained in this area since 1970 can be
traced to the following Theorem from a short two page paper by Leonard Carlitz
(Proc. Amer. Math. Soc. 11(1960), 391–392).
Theorem: The algebraic number field Z has class number ≤ 2 if and only if for
every nonzero integer α ∈ Z the number of primes πj in every factorization
α = π1π2 · · · πk
only depends on α.
While Carlitz’s result has been referenced well over 50 times, it did not initially
gain much attention. None the less, according to Math Science Net, since the
early 1970’s well over 200 papers dedicated to the study of factorization properties
of integral domains and monoids, and related topics, have appeared in refereed
mathematical journals or conference proceedings. During this same period, over 14
Doctoral Dissertations which touch on factorization properties have been completed.
The institutions where such degrees were awarded include The University of Iowa,
The University of Tennessee at Knoxville, The University of Nebraska at Lincoln,
The University of North Carolina at Chapel Hill, North Dakota State University,
Karl-Franzens-Universität-Graz and Universite D’Aix-Marseille III.
That this has become a highly active area of mathematical research is demon-
strated by Marcel Dekker’s publication in 1997 of the monograph Factorization in
Integral Domains, edited by D.D. Anderson. This monograph contains the Pro-
ceedings of a 1996 Mini-Conference held at the University of Iowa on factorization
problems followed by a Special Session on Commutative Algebra at the Midwestern
Regional Meeting of the American Mathematical Society. The meetings in Iowa
were followed by a flurry of research activity and the 2003 Meetings in Chapel Hill
xi

18. xii Preface
were organized with the intent of allowing some of the leading researchers in this
field to review the major results of this period.
I invited D. D. Anderson (University of Iowa), D. F. Anderson (University of
Tennessee at Knoxville), Jim Coykendall (North Dakota State University), Alfred
Geroldinger (Karl-Franzens-Universität, Graz), Franz Halter-Koch (Karl-Franzens-
Universität, Graz) and Ulrich Krause (Universität Bremen) to deliver 45 minute
addresses at the Mini-Conference. These authors have produced a series of 7 papers
based on these lectures which make up the introductory chapters of this volume.
These chapters demonstrate the diverse approach that many authors have taken in
studying nonunique factorizations. Several are written in a purely monoidal setting,
while several others demonstrate that ring theoretic properties play a key role in how
elements factor. Moreover, these chapters exhibit the broad range of mathematical
techniques necessary to obtain results in this area. Such techniques can range from
purely algebraic and combinatorial, to number theoretic. It is hoped that these
chapters not only offer mathematicians new to this area the opportunity to survey
its current trends and major results, but also offer an up to date introduction
to factorization theory for beginning graduate students. The remaining chapters
in this volume reflect research which is motivated by arithmetical properties of
commutative rings and monoids. This is not restricted to factorization problems,
as aspects of the following topics are all considered: multiplicative ideal theory, the
factorization of ideals and ideal generation problems, integer-valued polynomials,
Prüfer rings and Domains, block monoids and their combinatorial invariants, and
numerical monoids.
The two sessions in Chapel Hill and subsequent Proceedings Volume have gen-
erated a significant amount of interest. The sessions attracted 30 speakers from 6
different countries. This volume features 26 papers from 33 different contributors
from 7 different countries. I wish to take this opportunity to thank all those who
participated in the sessions, and to all those who submitted papers for consideration
to this volume. I am indebted to a large number of referees, who as always will
remain nameless. A special thanks goes to the Department of Mathematics at the
University of North Carolina at Chapel Hill, and in particular to Professor William
W. Smith who was instrumental in planning and organizing this event. I also wish
to thank the Department of Mathematics at Trinity University for their continued
support and help in preparing this manuscript. The organization and preparation
of the finished work would not have been possible without the help and patience
of Denise Wilson, our Departmental Secretary and technical typist. I am of course
indebted to Maria Allegra and her staff at Marcel Dekker for giving me the support
and opportunity to complete this manuscript.
I hope the readers of this volume enjoy its contents as much as I enjoyed putting
it together.
Scott Chapman
San Antonio, Texas
January 14, 2005

19. Contents
Preface xi
Contributors xv
1 Non-Atomic Unique Factorization in Integral Domains
Daniel D. Anderson 1
2 Divisibility Properties in Graded Integral Domains
David F. Anderson 22
3 Extensions of Half-Factorial Domains: A Survey
Jim Coykendall 46
4 C-Monoids and Congruence Monoids in Krull Domains
Franz Halter-Koch 71
5 Monotone Chains of Factorizations in C-Monoids
Andreas Foroutan and Alfred Geroldinger 99
6 Transfer Principles in the Theory of Non-unique Factorizations
Alfred Geroldinger and Franz Halter-Koch 114
7 Cale Monoids, Cale Domains, and Cale Varieties
Scott T. Chapman and Ulrich Krause 142
8 Weakly Krull Inside Factorial Domains
Daniel D. Anderson, Muhammed Zafrullah, and Gyu Whan Chang 172
9 The m-Complement of a Multiplicative Set
David F. Anderson and Gyu Whan Chang 180
10 Some Remarks on Infinite Products
Jim Coykendall 188
11 Rings with Prime Nilradical
Ayman Badawi and Thomas G. Lucas 198
xiii

20. xiv Contents
12 On the Ideal Generated by the Values of a Polynomial
Jean-Luc Chabert and Sabine Evrard 213
13 Using Factorizations to Prove a Partition Identity
David E. Dobbs and Timothy P. Kilbourn 226
14 On Inside Factorial Integral Domains
David E. Dobbs, Gabriel Picavet, and Martine Picavet-L’Hermitte 233
15 Polynomial Separation of Points in Algebras
Sophie Frisch 253
16 k-Factorized Elements in Telescopic Numerical Semigroups
Jose C. Rosales, Pedro A. Garcı́a-Sánchez, and Juan I. Garcı́a-Garcı́a 260
17 Prüfer Conditions in Rings with Zero-Divisors
Sarah Glaz 272
18 Unmixedness and the Generalized Principal Ideal Theorem
Tracy Dawn Hamilton 282
19 A Note on Sets of Lengths of Powers of Elements of Finitely
Generated Monoids
Wolfgang Hassler 293
20 UMV-Domains
Evan Houston and Muhammad Zafrullah 304
21 On Local Half-Factorial Orders
Florian Kainrath 316
22 On Factorization in Krull Domains with Divisor Class
Group Z2k
Karl M. Kattchee 325
23 Integral Morphisms
Jack Maney 337
24 A Special Type of Invertible Ideal
Stephen McAdam and Richard G. Swan 356
25 Factorization into Radical Ideals
Bruce Olberding 363
26 Strongly Primary Ideals
Gyu Whan Chang, Hoyoung Nam, and Jeanam Park 378
Index 389

21. Contributors
Daniel D. Anderson, Department of Mathematics, University of Iowa, Iowa City,
Iowa, 52242, dan-anderson@uiowa.edu
David F. Anderson, Department of Mathematics, University of Tennessee, Knoxville,
Tennessee 37996-1300, anderson@math.utk.edu
Ayman Badawi, Department of Mathematics and Statistics, American University of
Sharjah, P.O. Box 26666, Sharjah, United Arab Emirates, abadawi@avsharjah.edu
Jean-Luc Chabert, Laboratoire Amiénois de Mathématiques Fondamentale et Ap-
pliquée, CNRS-UMR 6140, Université de Picardie, 33 rue Saint Leu, 80039 Amiens,
France, jean-luc.chabert@u-picardie.fr
Gyu Whan Chang, Department of Mathematics, University of Incheon, Incheon,
402-748 Korea, whan@incheon.ac.kr
Scott T. Chapman, Department of Mathematics, Trinity University, One Trinity
Place, San Antonio, Texas 78212-7200, schapman@trinity.edu
Jim Coykendall, Department of Mathematics, North Dakota State University,Fargo,
North Dakota 58105-5075, Jim.Coykendall@ndsu.nodak.edu
David E. Dobbs, Department of Mathematics, University of Tennessee, Knoxville,
Tennessee 37996-1300, dobbs@math.utk.edu
Sabine Evrard, Laboratoire Amiénois de Mathématiques Fondamentale et Appliquée,
CNRS-UMR 6140, Université de Picardie, 33 rue Saint Leu, 80039 Amiens, France,
sabine.evrard@fnac.net
Andreas Foroutan, Institut für Mathematik, Karl-Franzens-Universität, Heinrich-
strasse 36, 8010 Graz, Austria, a−foroutan444@yahoo.com
Sophie Frisch, Institut für Mathematik, Technische Universität Graz, A-8010 Graz,
Austria, frisch@blah.math.tu-graz.ac.at
Pedro A. Garcı́a-Sánchez, Departamento de Álgebra, Universidad de Granada, Fac-
ultad de Ciencias, Campus Fuentenueva s/n, 18071 Granada, Spain, pedro@ugr.es
xv

22. xvi Contributors
Juan I. Garcı́a-Garcı́a, Departamento de Álgebra, Facultad de Ciencias, Universidad
de Granada, Campus Fuentenueva s/n, 18071 Granada, Spain, jigg@ugr.es
Alfred Geroldinger, Institut für Mathematik, Karl-Franzens-Universität, Heinrich-
strasse 36, 8010 Graz, Austria, alfred.geroldinger@uni-graz.at
Sarah Glaz, Department of Mathematics, University of Connecticut, Storrs, Con-
necticut 06269, glaz@uconnvm.uconn.edu
Franz Halter-Koch, Institut für Mathematik, Karl-Franzens-Universität, Heinrich-
strasse 36, 8010 Graz, Austria, franz.halterkoch@uni-graz.at
Tracy Dawn Hamilton, Department of Mathematics and Statistics, California State
University Sacramento, 6000 J Street, Sacramento, California 95819,
hamilton@csus.edu
Wolfgang Hassler, Institut für Mathematik, Karl-Franzens Universität Graz, Hein-
richstraße 36/IV, A-8010 Graz, Austria, wolfgang.hassler@uni-graz.at
Evan Houston, University of North Carolina at Charlotte, Department of Mathe-
matics, Charlotte, North Carolina 28223-0001, eghousto@email.uncc.edu
Florian Kainrath, Institut für Mathematik, Karl-Franzens-Universität Graz, Hein-
richstraße 36, A-8010 Graz, Austria, florian.kainrath@uni-graz.at
Karl M. Kattchee, Mathematics Department, University of Wisconsin-La Crosse,
1725 State Street, La Crosse, Wisconsin 54601, kattchee.karl@uwlax.edu
Ulrich Krause, Fachbereich Mathematik/Informatik, Universität Bremen, 28334
Bremen, Germany, krause@math.uni-bremen.de
Timothy P. Kilbourn, Department of Mathematics, University of Illinois at Urbana-
Champaign, Urbana, Illinois 61801-2975, tkilbour@math.uiuc.edu
Thomas G. Lucas, Department of Mathematics and Statistics, University of North
Carolina Charlotte, Charlotte, North Carolina 28223, tglucas@email.uncc.edu
Jack Maney, Department of Mathematics, North Dakota State University, Fargo,
North Dakota 58105-5075, Jack.L.Maney@ndsu.nodak.edu
Stephen McAdam, Department of Mathematics, The University of Texas at Austin,
1 University Station C1200, Austin, Texas 78712-0257, mcadam@math.utexas.edu
Hoyoung Nam, Department of Mathematics, Inha University, Incheon, 402-751,
Korea
Bruce Olberding, Department of Mathematical Sciences, New Mexico State Uni-
versity, Las Cruces, New Mexico 88003-8001, olberdin@NMSU.Edu

23. Contributors xvii
Jeanam Park, Department of Mathematics, Inha University, Incheon, 402-751, Ko-
rea
Gabriel Picavet, Laboratoire de Mathématiques Pures, Université Blaise Pascal,
63177 Aubière Cedex, France, Gabriel.Picavet@math.univ-bpclermont.fr
Martine Picavet-L’Hermitte, Laboratoire de Mathématiques Pures, Université Blaise
Pascal, 63177 Aubière Cedex, France, Martine.Picavet@math.univ-bpclermont.fr
Jose C. Rosales, Departamento de Álgebra, Universidad de Granada, Facultad de
Ciencias, Campus Fuentenueva s/n, 18071 Granada, Spain, jrosales@ugr.es
Richard G. Swan, Department of Mathematics, The University of Chicago, Chicago,
Illinois 60637, swan@math.uchicago.edu
Muhammed Zafrullah, Department of Mathematics, Idaho State University, Pocatello,
Idaho 83209-8085, mzafrullah@usa.net

25. Chapter 1
Non-Atomic Unique
Factorization in Integral
Domains
by Daniel D. Anderson
Abstract
UFDs can be characterized by the property that every nonzero nonunit is a
product of principal prime elements or equivalently that every nonzero nonunit
x can be written in the form x = upa1
1 · · · pan
n where u is a unit, p1, . . . , pn are
nonassociate principal primes, and each ai ≥ 1. Each pai
i , in addition to being
a power of a prime, has a number of other properties, each of which is subject
to generalization. We survey various generalizations of (unique) factorization
into prime powers in integral domains.
1 Introduction
Unique factorization domains are of course the integral domains in which every
nonzero nonunit element has a unique factorization (up to order and associates)
into irreducible elements or atoms. Now UFDs can also be characterized by the
property that every nonzero nonunit is a product of principal primes or equivalently
that every nonzero nonunit has the form upa1
1 · · · pan
n where u is a unit, p1, . . . , pn
are nonassociate principal primes, and each ai ≥ 1. Each of the pai
i , in addition
to being a power of a prime, has other properties, each of which is subject to
generalization. For example, each pai
i is primary, each is contained in a unique
maximal t-ideal, and the pai
i are pairwise coprime. The goal of this chapter is to
survey various generalizations of (unique) factorization into prime powers in integral
domains. This follows the thesis of M. Zafrullah that the pai
i are the building blocks
in a UFD. The author would like to thank M. Zafrullah for a number of discussions
of these topics over the past several years.
1

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