Non-Noetherian Commutative Ring Theory pp 117-137 | Cite as
On Generalized Lengths of Factorizations in Dedekind and Krull Domains
Abstract
The study of factorization in integral domains has played an important role in commutative algebra for many years. Much of this work has concentrated on the study of unique factorization domains (UFDs). Beyond the realm of UFDs, there is a large class of integral domains for which each nonunit can be factored as a product of irreducibles, yet the factorization may not be unique. Classically, Dedekind domains are such domains. A Dedekind domain D is a UFD if and only if its ideal class group is trivial. Hence the size of the class group becomes a measure of how far the domain D is from having “unique factorization”. The concept of the class group and its relation to factorization properties extends also to the more general class of Krull domains. It is within the context of these domains that we consider in this paper general questions concerning the lengths of factorization in the non-UFD setting.
Keywords
Prime Ideal Class Group Integral Domain Factorization Property Ideal ClassPreview
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References
- [1]D.D. Anderson, D.F. Anderson, S.T. Chapman and W.W. Smith, Rational elasticity of factorizations in Krull domains, Proc. Amer. Math. Soc. 117(1993), 37–43.MathSciNetMATHCrossRefGoogle Scholar
- [2]D.D. Anderson, D.F. Anderson and M. Zafrullah, Factorization in integral domains, J. Pure Appl. Algebra 69 (1990), 1–19.MathSciNetMATHCrossRefGoogle Scholar
- [3]D.D. Anderson, D.F. Anderson and M. Zafrullah, Factorization in integral domains, II, J. Algebra 152(1992), 78–93.MathSciNetMATHCrossRefGoogle Scholar
- [4]D.F. Anderson, Elasticity of factorizations in integral domains: a survey, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker 189(1997), 1–30.Google Scholar
- [5]D.F. Anderson, P.J. Cahen, S.T. Chapman and W.W. Smith, Some factorization properties of the ring of integer-valued polynomials, Lecture notes in Pure and Applied Mathematics, Marcel Dekker, volume 171(1995), chapter 10, 125–142.Google Scholar
- [6]D.F. Anderson, S.T. Chapman, F. Inman and W.W. Smith, Factorization in K[X 2 ,X 3 ] Arch. Math. 61(1993), 521–528.MathSciNetCrossRefGoogle Scholar
- [7]D.F. Anderson, S.T. Chapman and W.W. Smith, Some factorization properties of Krull domains with infinite cyclic divisor class group, J. Pure Appl. Algebra 96(1994), 97–112.MathSciNetMATHCrossRefGoogle Scholar
- [8]D.F. Anderson, S.T. Chapman and W.W. Smith, On Krull half-factorial domains with infinite cyclic divisor class group, Houston J. Math. 20(1994), 561–570.MathSciNetMATHGoogle Scholar
- [9]D.F. Anderson and P. Pruis, Length functions on integral domains, Proc. Amer. Math. Soc. 113(1991), 933–937.MathSciNetMATHCrossRefGoogle Scholar
- [10]L. Carlitz, A characterization of algebraic number fields with class number two, Proc. Amer. Math. Soc. 11(1960), 391–392.MathSciNetMATHGoogle Scholar
- [11]S. T. Chapman, On the Davenport constant, the cross number and their application in factorization theory, , Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, 189(1995), 167–190.Google Scholar
- [12]S. T. Chapman and J. Coykendall, Half-factorial Domains, A Survey, this volume.Google Scholar
- [13]S.T. Chapman, M. Freeze and W.W. Smith, Minimal zero-sequences and the strong Davenport constant, Discrete Math. 203(1999), 271–277.MathSciNetMATHCrossRefGoogle Scholar
- [14]S.T. Chapman and A. Geroldinger, On cross numbers of minimal zero-sequences, Australasian J. Comb. 14(1996), 85–92.MathSciNetMATHGoogle Scholar
- [15]S.T. Chapman and A. Geroldinger, Krull domains and monoids, their sets of lengths and associated combinatorial problems, Lecture notes in Pure and Applied Mathematics, Marcel-Dekker, 189(1997), 73–112.MathSciNetGoogle Scholar
- [16]S. T. Chapman and W.W. Smith, On a characterization of algebraic number fields with class number less than three, J. Algebra 135(1990), 381–387.MathSciNetMATHCrossRefGoogle Scholar
- [17]S.T. Chapman and W.W. Smith, Factorization in Dedekind domains with finite class group, Israel J. Math 71(1990), 65–95.MathSciNetMATHCrossRefGoogle Scholar
- [18]S. T. Chapman and W.W. Smith, On the k-HFD property in Dedekind domains with small class group, Mathematika 39(1992), 330–340.MathSciNetMATHCrossRefGoogle Scholar
- [19]S.T. Chapman and W.W. Smith, On the HFD, CHFD and fc-HFD properties in Dedekind domains, Comm. Algebra 20(1992), 1955–1987.MathSciNetMATHCrossRefGoogle Scholar
- [20]S.T. Chapman and W.W. Smith, An analysis using the Zaks-Skula constant of element factorizations in Dedekind domains, J. Algebra 159(1993), 176–190.MathSciNetMATHCrossRefGoogle Scholar
- [21]S.T. Chapman and W.W. Smith, On the length of factorizations of elements in an algebraic number ring, J. Number Theory 43(1993), 24–30.MathSciNetMATHCrossRefGoogle Scholar
- [22]S. T. Chapman and W.W. Smith, Finite cyclic groups and the fc-HFD property, Colloq. Math. 70(1996), 219–226.MathSciNetMATHGoogle Scholar
- [23]S.T. Chapman and W.W. Smith, Generalized sets of lengths, J. Algebra 200(1998), 449–471.MathSciNetMATHCrossRefGoogle Scholar
- [24]S. T. Chapman and W.W. Smith, An arithmetical characterization of finite elementary 2-groups, Comm. Algebra, to appear.Google Scholar
- [25]L. Claborn, Every abelian group is a class group”, Pac. J. Math. 18(1966), 219–222.MathSciNetMATHCrossRefGoogle Scholar
- [26]H. Cohn, A Second Course in Number Theory, J. Wiley & Sons (1962).Google Scholar
- [27]R. M. Fossum, The Divisor Class Group of a Krull Domain, Springer-Verlag, 1973.MATHCrossRefGoogle Scholar
- [28]M. Freeze, Lengths of Factorizations in Dedekind Domains, dissertation, The University of North Carolina at Chapel Hill, 1999.Google Scholar
- [29]W. D. Gao and A. Geroldinger, Systems of sets of lengths, II, Abhandl. Math. Sem. Univ. Hamburg, to appear.Google Scholar
- [30]W. D. Gao and A. Geroldinger, On long minimal zero sequences in finite abelian groups, Period. Math. Hungar., to appear.Google Scholar
- [31]A. Geroldinger, On non-unique factorizations in irreducible elements, II, Col. Math. Soc. Jânos Bolyai 51(1987), 723–752.MathSciNetGoogle Scholar
- [32]A. Geroldinger, Über nicht-eindeutige Zerlegunen in irreduzible Elements, Math. Z. 197(1988), 505–529.MathSciNetMATHCrossRefGoogle Scholar
- [33]A. Geroldinger, Systems von Langenmengen, Abh. Math. Sem. Univ. Hamburg 60(1990), 115–130.MathSciNetMATHCrossRefGoogle Scholar
- [34]A. Geroldinger, A structure theorem for sets of lengths, Coll. Math. 78(1998), 225–259.MathSciNetMATHGoogle Scholar
- [35]A. Geroldinger and F. Halter-Koch, On the asymptotic behaviour of lengths of factorizations, J. Pure Appl. Algebra 77(1992), 239–252.MathSciNetMATHCrossRefGoogle Scholar
- [36]R. Gilmer, Multiplicative Ideal Theory, Marcel Dekker, Inc., New York, 1972.MATHGoogle Scholar
- [37]R. Gilmer, W. Heinzer and W.W. Smith, On the distribution of prime ideals within the ideal class group, Houston J. Math. 22(1996), 51–59.MathSciNetMATHGoogle Scholar
- [38]A. Grams, The distribution of prime ideals of a Dedekind domain, Bull. Austral. Math. Soc. 11(1974), 429–441.MathSciNetCrossRefGoogle Scholar
- [39]F. Halter-Koch, On the asymptotic behaviour of the number of distinct factorizations into irreducibles, Ark. Mat. 31(1993), 297–305.MathSciNetMATHCrossRefGoogle Scholar
- [40]F. Halter-Koch, Elasticity of factorizations in atomic monoids and integral domains, J. Theorie des Nombres Bordeaux 7(1995), 367–385.MathSciNetMATHCrossRefGoogle Scholar
- [41]F. Halter-Koch, Finitely generated monoids, finitely primary monoids, and factorization properties of integral domains, Factorization in Integral Domains, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker 189(1997), 31–72.MathSciNetGoogle Scholar
- [42]H. B. Mann, Addition theorems: the addition theorems of group theory and number theory, Interscience Publishers, J. Wiley & Sons, 1965.MATHGoogle Scholar
- [43]D. Marcus, Number Fields, Springer-Verlag, New York, 1977.MATHCrossRefGoogle Scholar
- [44]W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, PWN-Polish Scientific Publishers, Warsaw, 1974.MATHGoogle Scholar
- [45]W. Narkiewicz, Some unsolved problems, Bull. Soc. Math France 25(1971), 159–164.MathSciNetMATHGoogle Scholar
- [46]M. B. Nathanson, Additive Number Theory, Springer-Verlag Graduate Texts in Mathematics 165(1996).CrossRefGoogle Scholar
- [47]J. Olson, A combinatorial problem on finite abelian groups I, J. Number Theory 1(1969), 8–11.MathSciNetMATHCrossRefGoogle Scholar
- [48]J. Olson, A combinatorial problem on finite abelian groups II, J. Number Theory 1(1969), 195–199.MathSciNetMATHCrossRefGoogle Scholar
- [49]A. Zaks, Half-factorial domains, Bull. Amer. Math. Soc. 82(1976), 721–3.MathSciNetMATHCrossRefGoogle Scholar
- [50]A. Zaks, Half-factorial domains, Israel J. Math. 37(1980), 281–302.MathSciNetMATHCrossRefGoogle Scholar