On the arithmetic of stable domains

Bashir, Aqsa; Geroldinger, Alfred; Reinhart, Andreas

doi:10.1080/00927872.2021.1929275

Cited by 6 publications

(5 citation statements)

References 50 publications

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“…Moreover, let R be a weakly Krull Mori domain. Then I * v (R) is transfer Krull if and only if it is halffactorial (in the local case this follows from Proposition 4.2.1 and from Proposition 4.9; the general case is a simple consequence, see [43,Proposition 7.3] and also [14,Theorem 5.9]). For more results of this flavor, see the references given in the discussion before Corollary 4.4).…”

Section: Transfer Krull Monoidsmentioning

confidence: 99%

“…and all these monoids are factorial. In orders of Dedekind domains with finite class group, monoids of all nonzero ideals and monoids of invertible ideals have similar arithmetical properties ( [16,38,14]). In contrast to (5.1) and in contrast to orders in Dedekind domains, our conjecture (Conjecture 5.12) is that the arithmetic of the monoid I(R) is completely different from the arithmetic of I * (R) and that it is as wild as it is for Krull monoids with infinite class group and prime divisors in all classes (see Proposition 4.9.3).…”

Section: On the Monoid Of Nonzero Ideals Of Polynomial Ringsmentioning

confidence: 99%

“…In Section 2, we introduce all the required arithmetical concepts in the setting of unit-cancellative monoids. The monoid of all nonzero ideals was studied for orders in Dedekind domains with finite class group [16,38,14]. In this setting, monoids of all nonzero ideals share arithmetical finiteness properties with monoids of invertible ideals and, more generally, with Krull monoids having finite class group.…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 5.8],[40, Theorem 5.8],[41, Corollary 4.6],[16, Theorem 1.1],[38, Theorem 5.13],[14, Theorem 5.10]). The following corollary characterizes when -under some additional assumptions -I * v (R) is fully elastic (compare with Proposition 4.9.1).…”

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confidence: 99%

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On the arithmetic of monoids of ideals

Geroldinger¹,

Khadam²

2021

Preprint

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We study the algebraic and arithmetic structure of monoids of invertible ideals (more precisely, of r-invertible r-ideals for certain ideal systems r) of Krull and weakly Krull Mori domains. We also investigate monoids of all nonzero ideals of polynomial rings with at least two indeterminates over noetherian domains. Among others, we show that they are not transfer Krull but they share several arithmetical phenomena with Krull monoids having infinite class group and prime divisors in all classes.

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Section: Transfer Krull Monoidsmentioning

confidence: 99%

Section: On the Monoid Of Nonzero Ideals Of Polynomial Ringsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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On the arithmetic of monoids of ideals

Geroldinger¹,

Khadam²

2021

Preprint

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“…The arithmetic of ideal semigroups in stable domains was recently studied by Bashir, Geroldinger, and Reinhart [BGR20]. In particular the semigroup of nonzero [invertible] ideals of R is transfer Krull if and only if it is half-factorial [BGR20, Theorem 5.10].…”

Section: Corollary 12 the Monoid T (R) Is Half-factorial If And Only ...mentioning

confidence: 99%

Lattices over Bass rings and graph agglomerations

Baeth,

Smertnig

2020

Preprint

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We study direct-sum decompositions of torsion-free, finitely generated modules over a (commutative) Bass ring R through the factorization theory of the corresponding monoid T (R). Results of Levy-Wiegand and Levy-Odenthal together with a study of the local case yield an explicit description of T (R). The monoid is typically neither factorial nor cancellative. Nevertheless, we construct a transfer homomorphism to a monoid of graph agglomerations-a natural class of monoids serving as combinatorial models for the factorization theory of T (R). As a consequence, the monoid T (R) is transfer Krull of finite type and several finiteness results on arithmetical invariants apply. We also establish results on the elasticity of T (R) and characterize when T (R) is half-factorial. (Factoriality, that is, torsionfree Krull-Remak-Schmidt-Azumaya, is characterized by a theorem of Levy-Odenthal.) The monoids of graph agglomerations introduced here may also be of independent interest to the factorization theory community.

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Apéry Sets and the Ideal Class Monoid of a Numerical Semigroup

Casabella,

D’Anna,

García-Sánchez

2023

Mediterr. J. Math.

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The aim of this article is to study the ideal class monoid $${\mathscr {C}}\ell (S)$$ C ℓ ( S ) of a numerical semigroup S introduced by V. Barucci and F. Khouja. We prove new bounds on the cardinality of $${\mathscr {C}}\ell (S)$$ C ℓ ( S ) . We observe that $${\mathscr {C}}\ell (S)$$ C ℓ ( S ) is isomorphic to the monoid of ideals of S whose smallest element is 0, which helps to relate $${\mathscr {C}}\ell (S)$$ C ℓ ( S ) to the Apéry sets and the Kunz coordinates of S. We study some combinatorial and algebraic properties of $${\mathscr {C}}\ell (S)$$ C ℓ ( S ) , including the reduction number of ideals, and the Hasse diagrams of $${\mathscr {C}}\ell (S)$$ C ℓ ( S ) with respect to inclusion and addition. From these diagrams, we can recover some notable invariants of the semigroup. Finally, we prove some results about irreducible elements, atoms, quarks, and primes of $$({\mathscr {C}}\ell (S),+)$$ ( C ℓ ( S ) , + ) . Idempotent ideals coincide with over-semigroups and idempotent quarks correspond to unitary extensions of the semigroup. We show that a numerical semigroup is irreducible if and only if $${\mathscr {C}}\ell (S)$$ C ℓ ( S ) has at most two quarks.

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On the arithmetic of stable domains

Cited by 6 publications

References 50 publications

On the arithmetic of monoids of ideals

On the arithmetic of monoids of ideals

Lattices over Bass rings and graph agglomerations

Apéry Sets and the Ideal Class Monoid of a Numerical Semigroup

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