BOOKS:

A. Geroldinger and F. Halter-Koch. Non-Unique Factorizations: Algebraic, Combinatorial and Analytic Theory, volume 278 of Pure and Applied Mathematics. Chapman & Hall/CRC, 2006.

PAPERS:

[1]     A. Geroldinger. Über nicht-eindeutige Zerlegungen in irreduzible Elemente. Math. Z., 197:505 - 529, 1988.

[2]     A. Geroldinger. Factorizations of algebraic numbers. In Number Theory, Ulm, volume 1380 of Lecture Notes in Math., pages 63 - 74. Springer, 1989.

[3]     A. Geroldinger and G. Lettl. Factorization problems in semigroups. Semigroup Forum, 40:23 - 38, 1990.

[4]     A. Geroldinger. Arithmetical characterizations of divisor class groups. Arch. Math., 54:455 - 464, 1990.

[5]     A. Geroldinger. Ein quantitatives Resultat über Faktorisierungen verschiedener Länge in algebraischen Zahlkörpern. Math. Z., 205:159 - 162, 1990.

[6]     A. Geroldinger. Systeme von Längenmengen. Abh. Math. Semin. Univ. Hamb., 60:115 - 130, 1990.

[7]     A. Geroldinger. On non-unique factorizations into irreducible elements. II. In Number Theory, Vol. II Budapest 1987, 

        volume 51 of Colloquia Mathematica      Societatis Janos Bolyai, pages 723 - 757. North Holland, 1990.

[8]     A. Geroldinger. Factorization of natural numbers in algebraic number fields. Acta Arith., 57:365 - 373, 1991.

[9]     A. Geroldinger. On the arithmetic of certain not integrally closed noetherian integral domains. Commun. Algebra, 19:685 - 698, 1991.

[10]   A. Geroldinger and F. Halter-Koch. Realization theorems for semigroups with divisor theory. Semigroup Forum, 44:229 - 237, 1992.

[11]   A. Geroldinger and F. Halter-Koch. On the asymptotic behaviour of lengths of factorizations. J. Pure Appl. Algebra, 77:239 - 252, 1992.

[12]   A. Geroldinger and F. Halter-Koch. Non-unique factorizations in block semigroups and arithmetical applications. Math. Slovaca, 42:641 - 661, 1992.

[13]   A. Geroldinger. Arithmetical characterizations of divisor class groups II. Acta Math. Univ. Comen., New Ser., 61:193 - 208, 1992.

[14]   A. Geroldinger and R. Schneider. On Davenport's constant. J. Comb. Theory, Ser. A, 61:147 - 152, 1992.

[15]   A. Geroldinger and J. Kaczorowski. Analytic and arithmetic theory of semigroups with divisor theory. J. Théor. Nombres Bordx., 4:199 - 238, 1992.

[16]   A. Geroldinger. On a conjecture of Kleitman and Lemke. J. Number Theory, 44:60 - 65, 1993.

[17]    A. Geroldinger. The complete integral closure of monoids and domains. PU.M.A., Pure Math. Appl., 4:147 - 165, 1993.

[18]    A. Geroldinger and F. Halter-Koch. Arithmetical theory of monoid homomorphisms. Semigroup Forum, 48:333 - 362, 1994.

[19]    A. Geroldinger. T-block monoids and their arithmetical applications to certain integral domains. Commun. Algebra, 22:1603 - 1615, 1994.

[20]    A. Geroldinger. The cross number of finite abelian groups. J. Number Theory, 48:219 - 223, 1994.

[21]    A. Geroldinger and R. Schneider. |The cross number of finite abelian groups II. Eur. J. Comb., 15:399 - 405, 1994.

[22]    A. Geroldinger, F. Halter-Koch, and J. Kaczorowski.Non-unique factorizations in orders of global fields. J. Reine Angew. Math., 459:89 - 118, 1995.

[23]    A. Geroldinger, F. Halter-Koch, and G. Lettl. The complete integral closure of monoids and domains II. Rend. Mat. Appl., VII. Ser., 15:281 - 292, 1995.

[24]    A. Geroldinger and J. Mockor. Quasi divisor theories and generalizations of Krull domains. J. Pure Appl. Algebra, 102:289 - 311, 1995.

[25]    A. Geroldinger.On the structure and arithmetic of finitely primary monoids. Czech. Math. J., 46:677 - 695, 1996.

[26]    A. Geroldinger and F. Halter-Koch. On quasi divisor theories and systems of valuations. Monatsh. Math., 121:55 - 67, 1996.

[27]    S.T. Chapman and A. Geroldinger. On cross numbers of minimal zero sequences. Australas. J. Comb., 14:85 - 92, 1996.

[28     A. Geroldinger and R. Schneider. The cross number of finite abelian groups III. Discrete Math., 150:123 - 130, 1996.

[29]    A. Geroldinger. Chains of factorizations in weakly Krull domains. Colloq. Math., 72:53 - 81, 1997.

[30]    A. Geroldinger. Chains of factorizations in orders of global fields. Colloq. Math., 72:83 - 102, 1997.

[31]    A. Geroldinger. Chains of factorizations and sets of lengths. J. Algebra, 188:331 - 363, 1997.

[32]    A. Geroldinger. The catenary degree and tameness of factorizations in weakly Krull domains. In Factorization in Integral Domains, 

         volume 189 of Lect. Notes Pure Appl. Math., pages 113 - 153. Marcel Dekker, 1997.

[33]    S.T. Chapman and A. Geroldinger. Krull domains and monoids, their sets of lengths and associated combinatorial problems.

          In Factorization in Integral Domains, volume 189 of Lect. Notes Pure Appl. Math., pages 73 - 112. Marcel Dekker, 1997.

[34]    A. Geroldinger and R. Schneider. On minimal zero sequences with large cross number. Ars Comb., 46:297 - 303, 1997.

[35]    A. Geroldinger. A structure theorem for sets of lengths. Colloq. Math., 78:225 - 259, 1998.

[36]    W. Gao and A. Geroldinger. On the structure of zerofree sequences. Combinatorica, 18:519 - 527, 1998.

[37]    W. Gao and A. Geroldinger. Half-factorial domains and half-factorial subsets in abelian groups. Houston J. Math., 24:593 - 611, 1998.

[38]    W. Gao and A. Geroldinger. On long minimal zero sequences in finite abelian groups. Period. Math. Hung., 38:179 - 211, 1999.

[39]    W. Gao and A. Geroldinger. Systems of sets of lengths II. Abh. Math. Semin. Univ. Hamb., 70:31 - 49, 2000.

[40]   G.A. Freiman and A. Geroldinger. An addition theorem and its arithmetical application. J. Number Theory, 85:59 - 73, 2000.

[41]   W. Gao and A. Geroldinger. On the order of elements in long minimal zero-sum sequences. Period. Math. Hung., 44:63 - 73, 2002.

[42]    A. Geroldinger and Y. ould Hamidoune. Zero-sumfree sequences in cyclic groups and some arithmetical application.J. Théor. Nombres Bordx., 14:221 - 239, 2002.

[43]   A. Geroldinger and F. Halter-Koch. Tamely generated ideals in finitary monoids. JP J. Algebra Number Theory Appl., 2:205 - 239, 2002.

[44]   A. Geroldinger, F. Halter-Koch, W. Hassler, and F. Kainrath. Finitary Monoids. Semigroup Forum, 67:1 - 21, 2003.

[45]   A. Geroldinger and R. Göbel. Half-factorial subsets in infinite abelian groups. Houston J. Math., 29:841 - 858, 2003.

[46]   W. Gao and A. Geroldinger. Zero-sum problems and coverings by proper cosets. Eur. J. Comb., 24:531 - 549, 2003.

[47]    W. Gao and A. Geroldinger. On zero-sum sequences in Z /n Z x Z /n Z. Integers, 3:Paper A08, 45p, 2003.

[48]    A. Geroldinger and F. Halter-Koch. Congruence monoids. Acta Arith., 112:263 - 296, 2004.

[49]    A. Geroldinger and F. Halter-Koch. Transfer principles in the theory of non-unique factorizations. In Arithmetical Properties of Commutative Rings and Monoids;volume 241 of Lecture Notes in Pure Appl. Math., pages 114 - 142. Chapman & Hall/CRC, 2005.

[50]    A. Foroutan and A. Geroldinger. Monotone chains of factorizations in C-monoids. In Arithmetical Properties of Commutative Rings and Monoids, 

         volume 241 of Lecture Notes in Pure Appl. Math., pages 99 - 113. Chapman & Hall/CRC, 2005.

[51]   W. Gao and A. Geroldinger. On a property of minimal zero-sum sequences and restricted sumsets. Bull. Lond. Math. Soc., 37:321 - 334, 2005.

[52]    R. Chi, S. Ding, W. Gao, A. Geroldinger, and W.A. Schmid. On zero-sum subsequences of restricted size IV. Acta Math. Hung., 107:337 - 344, 2005.

[53]    W. Gao and A. Geroldinger. On the number of subsequences with given sum of sequences over finite abelian p-groups. 

         Rocky Mt. J. Math., 37: 1541 -- 1550, 2007.

[54]     A. Geroldinger and F. Halter-Koch. Non-Unique Factorizations : a survey. In J.W. Brewer, S. Glaz, W. Heinzer, and B. Olberding, editors, 

          Multiplicative Ideal Theory in Commutative Algebra, 217 - 226, Springer 2006.

[55]    W. Gao and A. Geroldinger. Zero-sum problems in abelian groups : a survey. Expo. Math., 24:337 -- 369, 2006.

[56]    Y. Edel, C. Elsholtz, A. Geroldinger, S. Kubertin and L. Rackham. Zero-sum problems in finite abelian groups and affine caps. 

         Quarterly J. Math., Oxford II. Ser., 58:159 -- 186, 2007.

[57]    W. Gao, A. Geroldinger, and F. Halter-Koch. Group algebras of finite abelian groups and their applications to combinatorial problems, 

         Rocky Mt. J. Math.,  39:805 -- 823, 2009.

[58]    W. Gao, A. Geroldinger and W.A. Schmid. Inverse zero-sum problems. Acta Arith., 128:245 -- 279, 2007.

[59]    A. Geroldinger, W. Hassler and G. Lettl. On the arithmetic of strongly primary monoids. Semigroup Forum, 75: 567 -- 587, 2007.

[60]    A. Geroldinger and W. Hassler. Local tameness of v-noetherian monoids. J. Pure Appl. Algebra, 212:1509 -- 1524, 2008.

[61]    A. Geroldinger and W. Hassler. Arithmetic of Mori domains and monoids. J. Algebra, 319:3419 -- 3463, 2008.

[62]    M. Freeze and A. Geroldinger. Unions of sets of lengths. Funct. Approximatio, Comment. Math., 39:149 -- 162, 2008.

[63]    W. Gao and A. Geroldinger. On products of k-atoms. Monatsh. Math., 156: 141 -- 157, 2009.

[64]    A. Geroldinger. Additive group theory and non-unique factorizations. In A. Geroldinger and I. Ruzsa, Combinatorial Number Theory and Additive Group Theory,                    1 - 86, Advanced Courses in Mathematics, CRM Barcelona, Birkhäuser 2009. 

[65]    A. Geroldinger and D.J. Grynkiewicz. On the arithmetic of Krull monoids with finite Davenport constant. J. Algebra, 321:1256 -- 1284, 2009.

[66]    A. Geroldinger and D.J.  Grynkiewicz. On the structure of minimal zero-sum sequences with maximal cross number. 

         Journal of Combinatorics and Number  Theory, 1(2): 9 -- 26, 2009.

[67]    M. Freeze, W. Gao and A. Geroldinger. The critical number of finite abelian groups.  J. Number Theory, 129:2766 -- 2777, 2009.

[68]   W.Gao, A. Geroldinger and D. Grynkiewicz.  Inverse zero-sum problems III. Acta Arith., 141: 103 -- 152, 2010.

[69]    A. Geroldinger and F. Kainrath. On the arithmetic of tame monoids with applications to Krull monoids and Mori domains. 

         J. Pure Appl. Algebra, 214: 2199 -- 2218, 2010.

[70]   A. Geroldinger, D.J. Grynkiewicz, G.J. Schaeffer and W.A. Schmid. On the arithmetic of  Krull monoids with infinite cyclic class group.

        J. Pure Appl. Algebra, 214: 2219 -- 2250, 2010.

[71]   A. Geroldinger, D.J. Grynkiewicz and W.A. Schmid. The catenary degree of  Krull monoids I. 

        J. Theor. Nombres Bordx., 23:  137 -- 169, 2011.

[72]   A. Geroldinger, D.J. Grynkiewicz and W.A. Schmid. Zero-sum problems with congruence conditions. 

        Acta Math. Hung., 131:  323 -- 345, 2011.

[73]    A. Geroldinger,  M. Liebmann and A. Philipp. On the Davenport constant and on the structure of extremal zero-sum free sequences, Period. Math. Hung.,

          64: 213 -- 225, 2012

[74]    W.  Gao, A. Geroldinger and Qinghong Wang. A quantitative aspect of non-unique factorizations: the Narkiewicz constants.     

          Int. J. Number Theory, 7: 1463 -- 1502, 2011.

[75]    V.  Blanco, P.A. Garcia-Sanchez and A. Geroldinger.

          Semigroup theoretical characterizations of arithmetical invariants with applications to numerical monoids and Krull monoids.

Illinois Journal of Mathematics, 55: 1385 -- 1414

[76]    A. Geroldinger. Non-commutative Krull monoids: a divisor theoretic approach and their arithmetic. Osaka J. Mathematics, 50: 503 -- 539, 2013.

[77]    P. Baginski, A. Geroldinger, D.J. Grynkiewicz and A. Philipp.  Products of two atoms in Krull monoids and arithmetical  characterizations of class groups. Europ. J. Comb., 34: 1244 -- 1268, 2013.

[78]    A. Geroldinger and P. Yuan. The set of distances in Krull monoids. Bull. Lond. Math. Soc., 44:1203 -- 1208, 2012

[79]    A. Geroldinger and P. Yuan. The monotone catenary degree of Krull monoids. Results in Mathematics, 63: 999 -- 1031, 2013 .

[80]   A. Geroldinger and D.J. Grynkiewicz. The large Davenport constant I: groups with a cyclic index 2 subgroup. 

        J. Pure Appl. Algebra, 217:863 -- 885, 2013 .

[81]   W. Gao, A. Geroldinger and W.A. Schmid. Local and global tameness in Krull monoids. 

        Communications in Algebra, 43:262 -- 296, 2015.

[82]   N.R. Baeth and A. Geroldinger. Monoids of modules and arithmetic of direct-sum decompositions. 

        Pacific J. Math., 271:257 -- 319, 2014.

[83]   A. Geroldinger and F. Kainrath and A. Reinhart. Arithmetic of seminormal weakly Krull monoids and domains. 

        Journal of Algebra, 444:201 -- 245, 2015.

[84]    A. Geroldinger and S. Ramacher and A. Reinhart. On v-Marot Mori rings and C-rings. 

        Journal of the Korean Math. Soc., 52:1 -- 21, 2015.

[85]   N.R. Baeth and A. Geroldinger and D.J. Grynkiewicz and D.Smertnig. A semigroup theoretical view of direct-sum decompositions and associated combinatorial problems. 

        Journal of Algebra and its Appl., 14: 1550016 (60 pages), 2015.

[86]   A. Geroldinger and Qinghai Zhong. The set of minimal distances of Krull monoids. 

        Acta Arith., 173:97-120, 2016.

[87]   A. Geroldinger and Qinghai Zhong. The catenary degree of Krull monoids II. 

        Journal of the Australian Math. Soc., 98:324--354, 2015.

[88]   A. Geroldinger and W.A. Schmid. The system of sets of lengths in Krull monoids under set addition. 

        Revista Matematica Iberoamericana, 32:571 -- 588, 2016.

[89]   M. Freeze and W. Gao and A. Geroldinger. Corrigendum to "The critical number of finite abelian groups". 

        J. Number Theory, 152:205 -- 207, 2015.

[90]   A. Geroldinger and D.J. Grynkiewicz and P. Yuan. On products of k atoms II. 

        Moscow Journal of Combinatorics and Number Theory, 5:73 -- 129, 2015.

[91]   A. Geroldinger and W.A. Schmid. A characterization of class groups via sets of lengths. 

        (copyright: Korean Math. Soc.) J. Korean Math. Society, 56 (2019), 869 -- 915.

[92]   K. Cziszter and M. Domokos and A. Geroldinger. The interplay of invariant theory with multiplicative ideal theory and with arithmetic combinatorics. 

        in Multiplicative Ideal Theory and Factorization Theory, 43 -- 95,Springer 2016.

[93]   A. Geroldinger and Qinghai Zhong. A characterization of class groups via sets of lengths II. 

        J. Theor. Nombres Bordeaux, 29: 327 -- 346, 2017.

[94]   A. Geroldinger. Sets of lengths. 

        Amer. Math. Monthly, 123: 960 -- 988, 2016.

[95]   A. Geroldinger and Qinghai Zhong. The set of distances in seminormal weakly Krull monoids. 

        J. Pure Appl. Algebra, 220:3713 -- 3732, 2016.

[96]   Y. Fan and A. Geroldinger. Minimal relations and catenary degrees in Krull monoids. 

        J. Commutative Algebra, 11:29 -- 47, 2019.

[97]   A. Geroldinger and Wolfgang A. Schmid and Qinghai Zhong. Systems of sets of lengths: Transfer Krull monoids versus weakly Krull monoids. 

        in Rings, Polynomials, and Modules, 191 -- 235, Springer 2017.

[98]   A. Geroldinger and W.A. Schmid. A realization theorem for sets of distances. 

        J. Algebra, 481: 188 -- 198, 2017.

[99]   Y. Fan and A. Geroldinger and F. Kainrath and S. Tringali. Arithmetic of commutative semigroups with a focus on semigroups of ideals and modules. 

        Journal of Algebra and its Applications, 11: 1750234 (42 pages), 2017.

[100]   A. Geroldinger and E.D. Schwab. Sets of lengths in atomic unit-cancellative finitely presented monoids. 

        Colloquium Math., 151: 171 -- 187, 2018.

[101]   A. Geroldinger and Q. Zhong. Long sets of lengths with maximal elasticity. 

        Canadian J. Math., 70: 1284 -- 1318, 2018.

[102]   A. Geroldinger and W.A. Schmid. A realization theorem for sets of lengths in numerical monoids. 

        Forum Math., 30: 1111 -- 1118, 2018.

[103]   A. Geroldinger and A. Reinhart. The monotone catenary degree of monoids of ideals. 

        International Journal of Algebra and Computation, 29: 419 -- 457, 2019.

[104]   A. Geroldinger and M. Roitman. On strongly primary monoids and domains. 

        Communicatons in Algebra, 48: 4085 -- 4099, 2020.

[105]   A. Geroldinger and W.A. Schmid. Which sets are sets of lengths in all numerical monoids. 

        Banach Center Publications, 118: 181 -- 192, 2019.

[106]   A. Geroldinger and Q. Zhong. Sets of arithmetical invariants in transfer Krull monoids. 

      J. Pure Appl. Algebra, 223: 3889 -- 3918, 2019.

[107]   A. Geroldinger and Q. Zhong. A characterization of seminormal C-monoids. 

      Boll. Unione Mat. Ital., 12: 583 -- 597, 2019.

[108]   A. Geroldinger and W.A. Schmid. A characterization of Krull monoids for which sets of lengths are (almost) arithmetical progressions. 

      Revista Matematica Iberoamericana, 37: 293 -- 316, 2021.

[109]   J. Brantner and A. Geroldinger and A. Reinhart. On monoids of ideals of orders in quadratic number fields. 

      in Advances in Rings, Modules, and Factorizations, 11 -- 54, Springer 2020.

[110]   A. Geroldinger and Q. Zhong. Factorization theory in commutative monoids. 

      Semigroup Forum, 100: 22 -- 51, 2020.

[111]   A. Geroldinger and F. Gotti and S. Tringali. On strongly primary monoids with a focus on Puiseux monoids. 

      J. Algebra, 567: 310 -- 345, 2021.

[112]   A. Geroldinger and D.J. Grynkiewicz and J. Oh and Q. Zhong. On product-one sequences over dihedral groups. 

      Journal of Algebra and its Applications, 21: 2250064, 2022.

[113]   A. Geroldinger and W.A. Schmid. On the incomparability of systems of sets of lengths. 

      European Journal of Combinatorics, 111: 103694 (25 pages), 2023.

[114]   A. Bashir and A. Geroldinger and A. Reinhart. On the arithmetic of stable domains. 

      Commun. Algebra, 49: 4763 -- 4787, 2021.

[115]   A. Bashir and A. Geroldinger and Q. Zhong. On a zero-sum problem arising from factorization theory. 

      in Combinatorial and Additive Number Theory IV, 11 -- 24, Springer 2021.

[116]   A. Geroldinger and Q. Zhong. A realization result for systems of sets of lengths. 

      Israel Journal of Math., 247: 177 -- 193, 2022.

[117]   A. Geroldinger and Q. Zhong. A characterization of length-factorial Krull monoids. 

      New York J. Math., 27: 1347 -- 1374, 2021.

[118]   A. Geroldinger and F. Kainrath. On transfer homomorphisms of Krull monoids. 

      .Boll. Unione Mat. Ital., 14: 629 -- 646, 2021.

[119]   A. Geroldinger and M.Azeem Khadam. On the arithmetic of monoids of ideals. 

      Arkiv for Matematik, 60: 67 -- 106, 2022.

[120]   A. Geroldinger and F. Halter-Koch and Q. Zhong. On monoids of weighted zero-sum sequences and applications to norm monoids in Galois number fields and binary quadratic forms. 

      Acta Math. Hungarica, 168: 144 -- 185, 2022.

[121]   P.-Y. Bienvenu and A. Geroldinger. On algebraic properties of power monoids of numerical monoids. 

      Israel J. Math., to appear.

[122]   Gyu Whan Chang and A. Geroldinger. On Dedekind domains whose class groups are direct sums of cyclic groups. 

      J. Pure Appl. Algebra, 228: Article Id: 107470, 2024.

[123]    A. Geroldinger and Jun Seok Oh. On the isomorphism problem for monoids of product-one sequences. 

      submitted.

[124]    F. Fabsits and A. Geroldinger and A. Reinhart and Q. Zhong. On monoids of plus-minus weighted zero-sum sequences. the isomorphism problem and the characterization problem. 

      J. Commutative Algebra, 16: 1 -- 23, 2024.

[125]    A. Geroldinger and F. Gotti. On monoid algebras having every nonempty subset of $\mathbb{N}_{\ge 2}$ as a length set. 

      submitted.

[126]    A. Geroldinger and F. Kainrath. On Sets of Lengths in Monoids of Plus-Minus-Weighted Zero-Sum Sequences. 

      submitted.