The Art of Factorization in Multiplicative Structures

An MAA PREP Workshop held at Trinity University in San Antonio, Texas May 20th - May 25th, 2007

Reading List

This list is generally ordered with respect to the level of exposition.  We begin with a few papers that have appeared in “American Mathematical Monthly” type journals.  We follow this with a list of well written Undergraduate Research Projects, several of which led to publications in regular mathematical journals.  We offer a comprehensive list of expository writing – most of these papers appear in conference proceedings.  We follow this with two different lists of research level papers.  The first contains papers which include undergraduate co-authors.  The second is a list of “classic” papers which have been highly referenced in the literature.  We close with a few books which can be used as references.  When electronic versions of papers are available, we offer a link.

Some American Math Monthly type papers:

  1. S.T. Chapman, A simple example of non-unique factorization in integral domains, Amer. Math. Monthly 99(1992), 943-945.
  2. S.T. Chapman, Notes for the Participants of the 2006 Trinity REU Program on the Theory of Non-unique Factorizations.
  1. M. Banister, J. Chaika, S.T. Chapman and W. Meyerson, On a result of James and Niven concerning unique factorization in congruence semigroups, to appear in Elemente der Mathematik.

Some Well Written Undergraduate Research Projects:

  1. R. Kravitz, V. Noquez, and M. Sondjaja, Strongly Taut Restricted Block Monoids over Z, Trinity University REU Project 2006.
  1. G. Schaeffer, The Delta-set of a Singular ACM, Trinity University REU Project 2006.
  1. P. Baginski, C. Crutchfield, K. Kennedy, and M. Wright, Factorization and Elasticity with Applications to Rings of the Form K[x;S] and K[[x;S]], Trinity University REU Project 2005.
  1. K. Cervello, D. Terry, and L. Zhu, Extraction Degrees of Zero Sequences of Finite Abelian Groups, Trinity University REU Project 2004.
  1. C. Bowles, N. Kaplan, and D. Reiser, Delta Sets of Numerical Monoids, Trinity University REU Project 2004.
  1. B. McClain Factorisation Properties of Integer-Valued Polynomials, Trinity University Honors Thesis, 2004.
  1. M. Holden and T. Moore, Asymptotic Elasticity and the Full Elasticity Property in Atomic Monoids, Trinity University REU Project 2002.

Some Expository Papers:

  1. D.D. Anderson, GCD domains, Gauss’ lemma and contents of polynomials,    Non-Noetherian Commutative Ring Theory, Kluwer Academic Publishers, 2000, 1-32.
  1. D.D. Anderson, Non-atomic unique factorization in integral domains, Lecture Notes in Pure and Appl. Math. 241(2005), 1--21.
  1. D.F.Anderson, Elasticity of factorizations in integral domains: a survey, Lecture Notes in Pure and Appl. Math. 189(1997), 1--29.
  1. D.F. Anderson, Divisibility properties in graded integral domains, Lecture Notes in Pure and Appl. Math. 241(2005), 22-45.
  1. S.T. Chapman, On the Davenport constant, the cross number, and their application to factorization theory, Lecture Notes in Pure and Applied Mathematics 171(1995), 167-190.
  1. S.T. Chapman and J. Coykendall, Half-factorial domains: a survey, Non-Noetherian Commutative Ring Theory, Kluwer Academic Publishers, 2000, 97-116.
  1. S.T. Chapman, M. Freeze and W.W. Smith, On generalized lengths of factorizations in Dedekind and Krull domains, Non-Noetherian Commutative Ring Theory, Kluwer Academic Publishers, 2000, 117--138.
  1. S.T. Chapman and A. Geroldinger, Krull domains and monoids, their sets of lengths and associated combinatorial problems, Lecture Notes in Pure and Appl. Math. 189(1997), 73-112.
  1. J. Coykendall, Extensions of half-factorial domains: a survey, Lecture Notes in Pure and Appl. Math. 241(2005), 46--70.
  2. W. Gao and A. Geroldinger, Zero-sum problems in finite abelian groups: a survey, Expo. Math. 24(2006), 337-369.
  3. A. Geroldinger and F. Halter-Koch, Non-Unique Factorizations: a survey, Multiplicative Ideal Theory n Commutative Algebra, Springer, 2006, 207-226.
  4. F. Halter-Koch,  Non-unique factorizations of algebraic integers, preprint.
  5. W.A. Schmid: Half-factorial sets in finite abelian groups: A survey, Grazer Math. Ber., 348:41--64, 2005.

 

Some Research Papers with Undergraduate Co-authors:

  1. D. F. Anderson and S. Jenkens, Factorization in K[X^n,X^{n+1},…, X^{2n-1}], Comm. Algebra 23(1995), 2561--257.
  1. D.F. Anderson, S. T. Chapman, F. Inman, and W. W. Smith, Factorization in K[X^2,X^3], Arch. Math. (Basel) 61(1993), 521--528.
  1. D. F. Anderson and P. Pruis, Length functions on integral domains, Proc. Amer. Math. Soc. 113(1991), 933—937
  1. M. Banister, S.T. Chapman, J. Chaika, and W. Meyerson, On the arithmetic of arithmetical congruence monoids, to appear in Colloq. Math.
  1. C. Bowles, S. T. Chapman, N. Kaplan and D. Reiser, On Delta Sets of Numerical Monoids, J. Algebra Appl. 5(2006), 695--718.
  1.  S. T. Chapman, M. T. Holden and T. A. Moore, Full Elasticity In Atomic Monoids And Integral Domains, Rocky Mountain J. Math. 36(2006), 1437--1455.
  1. S.T. Chapman, J. Herr and N. Rooney, A factorization formula for class number two, J. Number Theory 79(1999), 58-66.
  1. S.T. Chapman, V. DeLorenzo and H. Swisher, On the asymptotic behavior of irreducibles in block semigroups, Semigroup Forum 63(2001), 34--48.

Some Classic Research Papers:

  1. D.D. Anderson, D.F. Anderson, and M. Zafrullah, Factorizations in integral domains, J. Pure Appl. Algebra 69 (1990), 1 – 19.
  1.  L. Carlitz, A characterization of algebraic number fields with class number two, Proc. Am. Math. Soc. 11 (1960), 391 – 392.
  1. J. Coykendall, On the integral closure of a half-factorial domain, J. Pure Appl. Algebra 180 (2003), 25 – 34.
  1. A. Geroldinger, Űber nicht-eindeutige Zerlegungen in irreduzible Elemente, Math. Z. 197 (1988), 505 – 529.
  1.  F. Halter-Koch, Elasticity of factorizations in atomic monoids and integral domains., J. Theor. Nombres Bordeaux 7(1995), 367--385.
  1. U. Krause, A characterization of algebraic number fields with cyclic class group of prime power order, Math. Z. 186 (1984), 143 – 148.
  1. M. Roitman, Polynomial extensions of atomic domains, J. Pure Appl. Algebra 87 (1993), 187 –199.  
  1. D.E. Rush, An arithmetic characterization of algebraic number fields with a given class group, Math. Proc. Camb. Philos. Soc. 94 (1983), 23 – 28.
  1. L. Skula, On c-semigroups, Acta Arith. 31 (1976), 247 – 257.
  1. A. Zaks, Half-factorial domains, Isr. J. Math. 37 (1980), 281 – 302.

Books

  1. A. Geroldinger and F. Halter-Koch, Non-unique Factorizations: Algebraic, Combinatorial and Analytic Theory, Pure and Applied Mathematics, 278, Chapman & Hall/CRC, 2006.
  1. W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, 3rd ed., Springer, 2004.

 

Questions can be directed to one of the organizers: schapman@trinity.edu or jim.coykendall@ndsu.edu