Project leader:
Alfred Geroldinger
FWF project number:
P 26036-N26


Alfred Geroldinger, Ao.Univ.-Prof. Mag. Dipl.-Ing. Dr.
+43 (0)316 380 - 5154
Heinrichstraße 36, Room 526 (4th floor)
Andreas Reinhart, Mag. Dr.rer.nat.
+43 (0)316 380 - 5155
Heinrichstraße 36, Room 528 (4th floor)
Daniel Smertnig, Dr.rer.nat. BSc MSc
+43 (0)316 380 - 5155
Heinrichstraße 36, Room 528 (4th floor)

All participants are located at the

Project summary

Non-Unique Factorizations. Let $R$ be a noetherian domain. Then every nonzero element of $R$ that is not a unit has a factorization into atoms (irreducible elements) of $R$. In general, there are many such decompositions, which differ not only up to units and the ordering of the factors. The main objective is to describe and classify the various phenomena of non-unique factorizations by arithmetical invariants and to relate these to the algebraic invariants of $R$. If $a = u_1 \cdots u_k$ is such a factorization of an element into atoms, then $k$ is called the length of the factorization, and the set $L(a)$ of all possible factorization lengths is the set of lengths of $a$. Sets of lengths are finite and nonempty, and they are central arithmetical invariants.

Multiplicative Ideal Theory. Its subject is the description of the multiplicative structure of an integral domain by means of ideals or certain systems of ideals (in technical terms, star and semistar operations, ideal and module systems). A main theme is the factorization of ideals (or of divisorial ideals, invertible ideals, and others) into prime ideals (or into radical ideals, primary ideals, and others).

Additive (Group and Number) Theory. The principal objects of study are the sumsets of (mainly finite and nonempty) subsets $A$ and $B$ of an additive abelian group. The most common reoccurring theme is the idea that if the sumset is small, then $A$, $B$ and their sumset must have structure. A further main object of study are subsequence sums and zero-sums of sequences over abelian groups, where a sequence (in this context) is a finite, unordered sequence allowing the repetition of elements. Problems dealing with sequences are often translated into problems with sets, and then they are studied via sumsets. The set of zero-sum sequences over a group (with concatenation as operation) is a Krull monoid.

This project is in the overlap of the above areas, and it is inspired by recent developments in them. Suppose that $R$ is integrally closed. Then $R$ is a Krull domain. Factoring elements is the same as factoring principal ideals, which lie inside the monoid of divisorial ideals. This gives rise to a transfer homomorphism from the elements of $R$ to the monoid of zero-sum sequences over the class group of $R$, which preserves sets of lengths. Via this machinery we study sets of lengths in $R$ with methods from Additive Theory. Furthermore, we study the ideal theory and sets of lengths in non-integrally closed domains.


  1. N. R. Baeth, A. Geroldinger, D. J. Grynkiewicz, and D. Smertnig,
    A semigroup-theoretical view of direct-sum decompositions and associated combinatorial problems.
    J. Algebra Appl., 14 (2015), no.2, 1550016, 60.
  2. N. R. Baeth and D. Smertnig,
    Arithmetical invariants of local quaternion orders.
  3. N. R. Baeth and D. Smertnig,
    Factorization theory: from commutative to noncommutative settings.
    J. Algebra, 441 (2015), 475–551.
  4. S. T. Chapman, M. Fontana, A. Geroldinger, and B. Olberding (editors),
    Multiplicative Ideal Theory and Factorization Theory.
    Proceedings in Mathematics and Statistics, vol. 107, Springer, 2016,
  5. K. Cziszter, 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.
  6. Y. Fan and A. Geroldinger,
    Minimal relations and catenary degrees in Krull monoids.
    J. Commut. Algebra. To appear.
  7. A. Garcia Elsener, P. Lampe, and D. Smertnig,
    Factoriality and class groups of cluster algebras.
  8. A. Geroldinger, D. J. Grynkiewicz, and P. Yuan,
    On products of k atoms II.
    Mosc. J. Comb. Number Theory, 5 (2015), no.3, 3–59.
  9. A. Geroldinger, F. Kainrath, and A. Reinhart,
    Arithmetic of seminormal weakly Krull monoids and domains.
    J. Algebra, 444 (2015), 201–245.
  10. A. Geroldinger, S. Ramacher, and A. Reinhart,
    On v-Marot Mori rings and C-rings.
    J. Korean Math. Soc., 52 (2015), no.1, 1–21.
  11. A. Geroldinger and A. Reinhart,
    The monotone catenary degree of monoids of ideals.
  12. A. Geroldinger and W. A. Schmid,
    A characterization of class groups via sets of lengths.
  13. A. Geroldinger and W. A. Schmid,
    A realization theorem for sets of distances.
    J. Algebra, 481 (2017), 188–198.
  14. A. Geroldinger and W. A. Schmid,
    The system of sets of lengths in Krull monoids under set addition.
    Rev. Mat. Iberoam., 32 (2016), no.2, 571–588.
  15. A. Geroldinger and Q. Zhong,
    The set of distances in seminormal weakly Krull monoids.
    J. Pure Appl. Algebra, 220 (2016), no.11, 3713–3732.
  16. O. A. Heubo-Kwegna, B. Olberding, and A. Reinhart,
    Group-theoretic and topological invariants of completely integrally closed Prüfer domains.
    J. Pure Appl. Algebra, 220 (2016), no.12, 3927–3947.
  17. A. Reinhart,
    A note on conductor ideals.
    Comm. Algebra, 44 (2016), no.10, 4243–4251.
  18. A. Reinhart,
    On the divisor-class group of monadic submonoids of rings of integer-valued polynomials.
    Commun. Korean Math. Soc., 32 (2017), no.2, 233–260.
  19. D. Smertnig,
    Every abelian group is the class group of a simple Dedekind domain.
    Trans. Amer. Math. Soc., 369 (2017), 2477–2491.
  20. D. Smertnig,
    Factorizations in bounded hereditary Noetherian prime rings.
    Proc. Edinburgh Math. Soc. (2). To appear.
  21. D. Smertnig,
    Factorizations of elements in noncommutative rings: A survey.
    In Multiplicative Ideal Theory and Factorization Theory, 353–402, Springer, 2016.