author image
Hello! I'm

Dr. Scott


Texas State University System Regents' Professor

Research interests: Commutative Algebra, Combinatorics, Number Theory

Refereed Papers

  1. “Integer-valued polynomials and almost division algorithms,” J. Nat. Sci. Math.28(1988),239-256.
  2. “On a characterization of algebraic number fields of class number less than three,” (with W.W.Smith) J. Algebra 135(1990), 361-366
  3. “Factorization in Dedekind domains with finite class group,” (with W.W. Smith) Israel J. Math.71(1990), 65-95.
  4. “Characterizing strong two-generators in K[x2, x3],” Houston J. Math. 16(1990), 217-229.
  5. “A Theorem on generating ideals in certain semigroup rings,” (with N. Vaughan) Boll. U.M.I.5-A(1991), 41-49.
  6. “On the HFD, CHFD, and k-HFD properties in Dedekind domains,” (with W.W. Smith) Comm.Algebra 20(1992), 1955-1987
  7. “A simple example of non-unique factorization in integral domains,” Amer. Math. Monthly99(1992), 943-945.
  8. “On the k-HFD property in Dedekind domains with small classgroup,” (with W.W. Smith) Math-ematika 39(1992), 330-340.
  9. “Rational elasticity of factorizations in Krull domains,” (with D.D. Anderson, D.F. Anderson, andW.W. Smith) Proc. Amer. Math. Soc. 117(1993), 37-43.
  10. “On the lengths of factorizations of elements in an algebraic number ring,” (with W.W. Smith)J. Number Theory 43(1993), 24-30.
  11. “An analysis using the Zaks-Skula constant of element factorizations in Dedekind domains,” (withW.W. Smith) J. Algebra 159(1993), 176-190.
  12. “Factorization in K[x2, x3],” (with D.F. Anderson, F. Inman*, and W.W. Smith) Arch. Math.(Basel) 61(1993), 521-528.
  13. “Some factorization properties of Krull domains with infinite cyclic divisor class group,” (withD.F. Anderson and W.W. Smith) J. Pure Appl. Algebra 96(1994), 97-112.
  14. “Overrings of half-factorial domains,” (with D.F. Anderson and W.W. Smith) Can. Math. Bull.37(1994), 437-442.“On Krull half-factorial domains with infinite cyclic divisor class group,” (with D.F. Andersonand W.W. Smith) Houston J. Math. 20(1994), 561-570.
  15. “On Krull half-factorial domains with infinite cyclic divisor class group,” (with D.F. Andersonand W.W. Smith) Houston J. Math. 20(1994), 561-570.
  16. “Some factorization properties of the ring of integer-valued polynomials,” (with D.F. Anderson,P-J. Cahen and W.W. Smith) Lecture Notes in Pure and Applied Mathematics, Marcel Dekker,171(1995), 125-142.
  17. “On the Davenport constant, the cross number, and their application to factorization theory,”Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, 171(1995), 167-190.
  18. “Overrings of half-factorial domains, II,” (with D.F. Anderson and W.W. Smith) Comm. Algebra23(1995), 3961-3976.
  19. “Factorization sets and half-factorial sets in integral domains,” (with D.F. Anderson and W.W.Smith) J. Algebra 178(1995), 92-121.
  20. “Finite cyclic groups and the k-HFD property,” (with W.W. Smith) Colloq. Math.70(1996),219-226.
  21. “On a generalization of a theorem of Zaks and Skula,” (with W. Thill*) Proc. Royal Irish Aca.96A(1996), 79-83.
  22. “On cross numbers of minimal zero sequences,” (with A. Geroldinger) Australasian J. Comb.14(1996), 85-92.
  23. “Two classes of ideals determined by integer-valued polynomials,” (with P-J. Cahen, K. Roegnerand W.W. Smith) Rend. Mat. (Roma) 16(1996), 625-636.
  24. “A basis for the ring of polynomials integer-valued on prime numbers,” (with J-L. Chabertand W.W. Smith) Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, 189(1997),271-284.
  25. “Krull domains and monoids, their sets of length and associated combinatorial problems,” (withA. Geroldinger) Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, 189(1997),73-112.
  26. “Algebraic properties of the ring of polynomials integer-valued on prime numbers,” (with J-L.Chabert and W.W. Smith) Comm. Algebra 25(1997), 1945-1959.
  27. “Generalized sets of length,” (with W.W. Smith) J. Algebra 200(1998), 449-471.
  28. “Integral domains with highly non-unique factorization,” (with D.D. Anderson and J. Kwak)Results Math. 33(1998), 22-29.
  29. “Criteria for unique factorization,” (with D.D. Anderson, F. Halter-Koch, and M. Zafrullah) J.Pure Appl. Algebra 127(1998), 205-218.
  30. “Generators of maximal ideals in the ring of integer-valued polynomials,” (with W.W. Smith)Rocky Mountain J. Math. 28(1998), 95-105.
  31. “The Skolem property in rings of integer-valued polynomials,” (with J-L. Chabert and W.W.Smith) Proc. Amer. Math. Soc. 126(1998), 3151-3159.
  32. “Minimal zero-sequences and the strong Davenport constant,” (with M. Freeze and W.W. Smith)Discrete Math. 203(1999), 271-277.
  33. “A factorization formula for class number two,” (with J. Herr* and N. Rooney*) J. NumberTheory 79(1999), 58-66.
  34. “On the elasticities of Krull domains with finite cyclic divisor class group,” (with D.F. Anderson)Comm. Algebra 28(2000), 2543-2553.
  35. “Monoids determined by a homogenous linear Diophantine equation and the half-factorial prop-erty” (with U. Krause and E. Oeljeklaus) J. Pure Appl. Algebra 151(2000), 107-133.
  36. “On generalized lengths of factorizations in Dedekind and Krull domains,” (with M. Freeze and W.W. Smith) Non-Noetherian Commutative Ring Theory, Kluwer Academic Publishers, 117-138.
  37. “Half-factorial domains, a survey,” (with J. Coykendall) Non-Noetherian Commutative RingTheory, Kluwer Academic Publishers, 97-116.
  38. “100 Problems in commutative ring theory,” (with S. Glaz) Non-Noetherian Commutative Ring Theory, Kluwer Academic Publishers, 459-476.
  39. “An arithmetical characterization of finite elementary 2-groups,” (with W.W. Smith) Comm.Algebra 29(2001), 1249-1257.
  40. “Equivalence classes of minimal zero-sequences modulo a prime,” (with M. Freeze and W.W.Smith) Lecture Notes in Pure and Applied Mathematics Marcel-Dekker, 220(2001), 133-146.
  41. “On the asymptotic behavior of irreducibles in block semigroups,” (with V. DeLorenzo* and H.Swisher*) Semigroup Forum 63(2001), 34-48.
  42. “Sets of lengths in V + XB[X] domains,” (with N. Gonzalez and S. Pellerin) Arabian J. Sci.Engrg. 26(2001), 69-82.
  43. “Computing the elasticity of a Krull monoid,” (with J.I. Garcıa-Garcıa, P.A. Garc´ıa-Sánchez andJ.C. Rosales) Linear Algebra Appl. 336(2001), 201-210
  44. “The strong two-generator property in rings of integer-valued polynomials determined by finitesets,” (with A. Loper and W.W. Smith) Arch. Math. (Basel) 78(2002), 372-377.
  45. “On Davenport’s constant of finite abelian groups,” (with M. Freeze, W. D. Gao, and W. W.Smith) Far East J. Math. Sci. 5 (2002), no. 1, 47-54.
  46. “On Diophantine monoids and their class groups,” (with U.Krause and E. Oeljeklaus) Pacific J.Math. 207(2002), 125-147.
  47. “Inside factorial monoids and integral domains,” (with F. Halter-Koch and U. Krause) J. Algebra252(2002), 350-375.
  48. “On the number of factorizations of an element in an atomic monoid,” (with J.I. Garcıa-Garcıa,P.A. Garc´ıa-Sánchez and J.C. Rosales) Adv. in Appl. Math. 29(2002), 438-453.
  49. “On the asymptotic values of length functions in Krull and finitely generated commutativemonoids,” (with J.C. Rosales) J. Aust. Math. Soc. 74(2003), 421-436.
  50. “On Factorization in Block Monoids formed by {1, a} in Zn,” (with W.W. Smith) Proc. EdinburghMath. Soc. 46(2003), 257-267
  51. “On cross numbers of minimal zero sequences in certain cyclic groups,” (with P. Baginski*, K.McDonald* and L. Pudwell*) Ars Combin. 70(2004), 47-60.
  52. “Strongly two-generated ideals in rings of integer-valued polynomials determined by finite sets,”(with A. Loper and W. W. Smith) C.R. Math. Rep. Acad. Sci. Canada 26(2004), 33-38.
  53. “Cale monoids, Cale domains and Cale varieties,” (with U.Krause) Arithmetical Properties ofCommutative Rings andMonoids, Lecture Notes in Pure and Applied Mathematics 241(2005),142-171.
  54. “Irreducible polynomials and full elasticity in rings of integer-valued polynomials,” (with B.McClain*) J. Algebra 293(2005), 595-610.
  55. “A characterization of minimal zero-sequences of index one in finite cyclic groups,” (with W. W.Smith) Integers 5(1)(2005), #A27.
  56. “Asymptotic elasticity in atomic monoids,” (with P. Baginski, M. Holden* and T. Moore*)Semigroup Forum 72(2006), 134-142.
  57. “Restricted elasticity and rings of integer-valued polynomials determined by finite subsets,” (withW. W. Smith) Monatsh. Math. 148(2006), 195-203.
  58. “The catenary and tame degree in finitely generated commutative cancellative monoids,” (with P.A. Garc´ıa-Sánchez, D. Llena, V. Ponomarenko and J. C. Rosales) Manuscripta Math. 120(2006),253-264.
  59. “Presentations of finitely generated cancellative commutative monoids and nonnegative solutionsof systems of linear equations,” (with P. A. Garc´ıa-Sánchez, D. Llena and J. C. Rosales) DiscreteAppl. Math. 154(2006), 1947-1959.
  60. “On full elasticity in atomic monoids and integral domains,” (with M. Holden* and T. Moore*) Rocky Mountain J. Math.36(2006), 1437-1455.
  61. “On Delta sets of numerical monoids,” (with C. Bowles*, N. Kaplan* and D. Reiser*) JournalAlgebra Appl. 5(2006), 1-24.
  62. “Robert Gilmer’s contributions to the theory of integer-valued polynomials,” (with V. Pono-marenko and W. W. Smith) Multiplicative Ideal Theory in Commutative Algebra, a tribute to thework of Robert Gilmer, Springer, New York, 2006,109-122.
  63. “Elastic properties and prime elements,” (with P. Baginski, C. Crutchfield*, K. Kennedy* andM. Wright*) Results Math. 49(2006), 1-14.
  64. “Presentations of finitely generated cancellative monoids and natural solutions of linear systemsof equations,” (with P. A. Garc´ıa-Sánchez, D. Llena and J. C. Rosales) Fifth Conference onDiscrete Mathematics and Computer Science, Ciencias (Valladolid), Univ. Valladolid, Secr.Publ. Intercamb. Ed. 23(2006), 217-224.
  65. “On the arithmetic of arithmetical congruence monoids,” (with M. Banister*, J. Chaika* and W.Meyerson*) Colloq. Math. 108(2007), 105-118.
  66. “On a result of James and Niven concerning unique factorization in congruence semigroups,”(with M. Banister*, J.Chaika* and W. Meyerson*) Elem. Math 62(2007),68-72.
  67. “On minimum delta set values in block monoids over cyclic groups,” (with S. Chang and W. W.Smith) Ramanujan J. 14(2007), 155-171
  68. “An inequality concerning the elasticity of of Krull monoids with divisor class group Zp,” (withW. W. Smith) Ric. Mat. 56(2007), 107-117.
  69. “Elasticity in certain block monoids via the euclidean table,” (with S. Chang and W. W. Smith)Math. Slovaca 57(2007), 415-454.
  70. “Sets of lengths do not characterize numerical monoids,” (with J. Amos*, N. Hine* and J.Paixao*) Integers 7(2007), #A50.
  71. “On the delta-set of a singular arithmetical congruence monoid,” (with P. Baginski and G.Schaeffer*) J. Théor. Nombres Bordeaux 20(2008), 45-59.
  72. “On the asymptotic behavior of unions of sets of lengths in atomic monoids,” (with P. Baginski,N. Hine* and J. Paixao*) Involve 1(2008), 101-110.
  73. “On minimal distances in Krull monoids with infinite classgroup,” (with W. A. Schmid and W.W. Smith) Bull. Lond. Math. Soc 40(2008), 613-618.
  74. “Quasi-half-factorial subsets of abelian torsion groups,”(with W. A. Schmid and W. W. Smith)Houston J. Math. 35(2009), 13-22.
  75. “The catenary and tame degree of numerical semigroups,” (with P. A. Garc´ıa-Sánchez and D.Llena) Forum Math 21(2009), 117-129.
  76. “A theorem on accepted elasticity in certain local arithmetical congruence monoids,” (with M.Banister*, J. Chaika* and W. Meyerson*) Abh. Math. Sem. Univ. Hamburg 79(2009), 79-86.
  77. “Delta sets of numerical monoids are eventually periodic,” (with R. Hoyer* and N. Kaplan)Aequationes Math. 77(2009), 273-279.
  78. “Elastic properties of some semirings defined by positive systems,” (with P. Cesarz*, S. McAdamand G. Schaeffer) Proceedings of the Fifth International Fez Conference on Commutative Algebraand its Applications, Walter de Gruyter, Berlin, 2009, 89-101.
  79. “A note on the sum of the quotient and remainder in the division algorithm,” (with D. F.Anderson and W. W. Smith) Elem. Math 64(2009), 154-159.
  80. “A note on cancellation properties of semistar operations,” (with R. Matsuda) Int. Elect. J.Algebra 7(2010), 78-86.
  81. “Delta sets of numerical monoids using non-minimal sets of generators,” (with G. Daigle*, R.Hoyer* and N. Kaplan) Comm. Algebra. 38(2010), 2622-2634.
  82. “On the delta set and catenary degree of Krull Monoids with infinite cyclic divisor class group,”(with P. Baginski, R. Rodriguez*, G. Schaeffer and Y. She*) J. Pure Appl. Algebra. 214(2010),1334-1339.
  83. “How far is an element from being prime?” (with D. F. Anderson) Journal Algebra Appl. 9(2010),1-11.
  84. “Elasticity in generalized arithmetical congruence monoids,” (with D. Steinberg*) Results Math.58(2010), 221-231.
  85. “Elements in a numerical semigroup with factorizations of the same length,” (with P. A. Garcıa-Sánchez, D. Llena and J. Marshall*) Canad. Math. Bull. 54(2011), 39-43.
  86. “An algorithm to compute ω-primality in a numericalmonoid,” (with D. F. Anderson, N. Kaplanand D. Torkornoo*) Semigroup Forum 82(2011), 96-108.
  87. “Factorizations of algebraic integers, block monoids and additive number theory,” (with P. Ba-ginski) Amer. Math. Monthly 118(2011), 901-920.
  88. “On image sets of integer-valued polynomials,” (with V. Ponomarenko) J. Algebra 348(2011),350-353
  89. “A closer look at non-unique factorization via atomic decay and strong atoms,” (with U. Krause)Progress in Commutative Algebra, 2, Walter de Gruyter & Co., Berlin, Germany, 2012, 301-315.
  90. “On the Delta sets and minimal presentations of a finitely generated commutative monoid,” (withP.A. Garc´ıa-Sánchez, D. Llena, A. Malyshev and D. Steinberg) Arabian J. Math. 1(2012), 53-61.
  91. “Erdös-Zaks all divisor sets,” (with W. W. Smith) Period. Math. Hungar. 64(2012), 227-246.
  92. “On bounding measures of primeness in integral domains,” (with D. F. Anderson) Internat. J.Algebra Comput. 22(2012), 15 pp
  93. “Algebraic and combinatorial aspects of sandpile monoids on directed graphs,” (with L.Garcia-Puente, R. Garcia, M. Malandro, K. Smith) J. Comb. Theory Ser. A. 120(2013), 245-265.
  94. “Shifts of generators and delta sets of numerical monoids,” (with N. Kaplan, T. Lemburg*, A.Niles* and C. Zlogar*) Internat. J. Algebra Comput. 24(2014), 655-669.
  95. “A tale of two monoids: a friendly introduction to the theory of non-unique factorizations,” Math.Magazine 87(2014), 163-173.
  96. “Arithmetic congruence monoids: A Survey,” (with P. Baginski) Combinatorial and AdditiveNumber Theory: CANT 2011 and 2012, Springer Proceedings in Mathematics and Statistics,101(2014).
  97. “On the omega values of generators of embedding dimension three numerical monoids generatedby an interval,” (with W. Puckett* and K. Shour*) Involve 7(2014), 657-667.
  98. “On Delta Sets and their Realizable Subsets in Krull Monoids with Cyclic Class Groups,” (withF. Gotti* and R. Pelayo) Colloq. Math. 137(2014), 137-146.
  99. “The catenary and tame degrees on a numerical monoid are eventually periodic,” (with M.Corrales*, A. Miller*, C. Miller*, and D. Patel*) J. Aust. Math. Soc. 97(2014), 289-300.
  100. “I prefer pi: A brief history and anthology of articles in the American Mathematical Monthly,”(with J. Borwein) Amer. Math. Monthly 122(2015), 195-216.
  101. “I prefer pi: Addendum,” Amer. Math. Monthly 122(2015), 800.
  102. “ω-primality in embedding dimension three numerical semigroups,” (with P. A. Garc´ıa-Sánchez,Z. Tripp*, C. Viola) J. Algebra Appl. 63(2016).
  103. “The catenary degree of elements in numerical monoids generated by arithmetic sequences,”(with M. Corrales*, A. Miller*, C. Miller*, and D. Patel*) Comm. Algebra 45(2017), 5443-5452.
  104. “What happens when the division algorithm “almost” works,” Amer. Math. Monthly 125(2018),643-647.
  105. “Factorization in the Chicken McNugget monoid,” (with C. O’Neill) Math. Magazine 91(2018),323-336.
  106. “So what is class number 2?” Amer. Math. Monthly 126(2019), 330-339.
  107. How do elements really factor in Z[√−5],” (with F. Gotti and M. Gotti) Advances in Commu-tative Algebra Dedicated to David F. Anderson, Birkhäuser, 2019, 171-196.
  108. “ω-primality in arithmetic Leamer monoids,” (with Z. Tripp) Semigroup Forum 99(2019), 47-56.
  109. “Factorization invariants of Puiseux monoids generated by geometric sequences,” (with F. Gottiand M. Gotti) Comm. Algebra 48(2020), 380-396.
  110. “Beyond coins, stamps, and Chicken McNuggets: an invitation to numerical semigroups,” (withR. Garcia and C. O’Neill) A Project Based Guide in Undergraduate Mathematics: Starting andSustaining Undergraduate Research, Birkhäuser, 2020, 177-202.
  111. “When is a Puiseux monoid atomic?,” (with F. Gotti and M. Gotti) to appear in Amer. Math.Monthly.
  112. “Finer factorization characterizations of class number 2,” to appear in Rocky Mountain J. Math.
  113. “Distances between factorizations in the Chicken McNugget monoid,” (with C. O’Neill and P.A. Garc´ıa-Sánchez) to appear in College Math. Journal.
  114. “Length-factoriality in commutative monoids and integral domains,” (with J. Coykendall, F.Gotti, and W. W. Smith) to appear in Journal of Algebra.
  115. “Bi-atomic classes of positive semirings” (with N. Baeth and F. Gotti) to appear in SemigroupForum.

Papers Submitted or in Progress

  1. “On length Densities,” (with C. O’Neill and V. Ponomarenko) submitted.
  2. “On the atomicity of positive monoids and semirings,” (with F. Gotti and M. Gotti) in preparation.
  3. The importance of being prime, a nontrivial generalization for nonunique factorizations,” (withN. Baeth) in preparation.
  4. “Generating ideals in subrings ofK[[X]] via numerical semigroups,” in preparation.

Professional Nontechnical Writing

  1. “REU Spotlight: Trinity University,” Math Horizons 11(2004), pp. 26-27.
  2. “The Trinity University Research Experiences for Undergraduates in Mathematics Program,”Proceedings of the Conference on Promoting Undergraduate Research in Mathematics, AmericanMathematical Society, Providence, RI, 2007.
  3. “Remember the Alamonoid: The Art of Factorization in Multiplicative Systems in San Antonio,”Focus 27(2007), pp. 19-20.
  4. “A letter from the editor,” Amer. Math. Monthly 119(2012), 3.
  5. “Editor’s Endnotes,” Amer. Math. Monthly 119(2012), 892-894.
  6. “A letter from the editor,” Amer. Math. Monthly 120(2013), 383.
  7. “Editor’s Endnotes,” Amer. Math. Monthly 120(2013), 482-483.
  8. “Editor’s Endnotes,” Amer. Math. Monthly 120(2013), 954-956.
  9. “A letter from the editor: What’s in a color?” Amer. Math. Monthly 121(2014), 3-4.
  10. “Editor’s Endnotes,” Amer. Math. Monthly 121(2014), 467-468.
  11. “Editor’s Endnotes,” Amer. Math. Monthly 121(2014), 957-959.
  12. “A letter from the editor:2015 is the centennial year of the MAA,” Amer. Math. Monthly122(2015), 3.
  13. “Editor’s Endnotes,” Amer. Math. Monthly 122(2015), 511-512.
  14. “Editor’s Endnotes,” Amer. Math. Monthly 122(2015), 1023-1025.
  15. “How to deal with a Mathematics journal Editor,” Notices Amer. Math. Soc. 63(2016), 182-184.
  16. “Foreword,” Multiplicative Ideal Theory and Factorization Theory - Commutative and Non-Commutative Perspectives, Proceedings in Mathematics and Statistics 170(2016), Springer In-ternational, vii-xiv.
  17. “Editor’s Endnotes,” Amer. Math. Monthly 123(2016), 515-516.
  18. “A letter from the editor: Jonathan M. Borwein 1951-2016,” Amer. Math. Monthly 123(2016),847-848.
  19. “Single digit NFL scores,” (published under a psuedonym) Amer. Math. Monthly 123(2016),959.
  20. “A letter from the editor: The long and winding road,” Amer. Math. Monthly 123(2016),955-958.
  21. “From the guest editors,” (with J. Gallian) Amer. Math. Monthly 124(2017), 771.
  22. “Ethics lessons learned while editing the Monthly: Modern publishing is raising new issues,”Notices Amer. Math. Soc. 65(2018), 1270-1272.

Papers in Translation

  1. “How to deal with a Mathematics journal Editor,” Vietnamese translation,Newsletter of theVietnamese Mathematical Society3(2017), 10–14.
  2. "I prefer pi: A brief history and anthology of articles in the American Mathematical Monthly, I”(with J. Borwein) Chinese translation,Mathematics3(2017), 207–218.
  3. “I prefer pi: A brief history and anthology of articles in the American Mathematical Monthly, II”(with J. Borwein) Chinese translation,Mathematics4(2017), 311–319.


  1. Non-Noetherian Commutative Ring Theory, (edited with S. Glaz) Kluwer Academic Publishers,Boston, 2000.
  2. Arithmetical Properties of Commutative Rings and Monoids(edited), Lecture Notes in Pure andApplied Mathematics,241(2005) Chapman and Hall, New York.
  3. Multiplicative Ideal Theory and Factorization Theory - Commutative and Non-Commutative Per-spectives(edited with M. Fontana, A. Geroldinger, and B. Olberding), Proceedings in Mathe-matics and Statistics170(2016), Springer International.
  4. Numerical Semigroups, IMNS 2018, (edited with V. Barucci, M. D’ana, R. Froberg) SpringerINdAM Series, 2020.
Back to top