Lattices over Bass Rings and Graph Agglomerations

Baeth, Nicholas R.; Smertnig, Daniel

doi:10.1007/s10468-021-10040-2

Cited by 13 publications

(8 citation statements)

References 47 publications

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“…4] for details. Another motivation comes from the monoid-theoretic approach to the study of direct sum decompositions of modules pioneered by A. Facchini and R. Wiegand, see [13,30,7,4,6] and references therein (we will come back to this point in Sect. 4.3).…”

Section: Applicationsmentioning

confidence: 99%

An Abstract Factorization Theorem and Some Applications

Tringali

2021

Preprint

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We combine the language of monoids with the language of preorders to formulate an abstract factorization theorem with several applications. In particular, this leads to (i) a generalization of P. M. Cohn's classical theorem on "atomic factorizations" from cancellative to Dedekind-finite monoids (and, hence, to a variety of rings that are not domains); (ii) a monoid-theoretic proof that every module of finite uniform dimension over a (commutative or non-commutative) ring R is isomorphic to a direct sum of finitely many indecomposable R-modules (in fact, we obtain the result as a special case of a general decomposition theorem for the objects of certain categories with finite products, where the indecomposable R-modules are characterized as the atoms of a certain "monoid of modules"). Also, we recover and extend an existence theorem of D. D. Anderson and S. Valdes-Leon on "irreducible factorizations" in commutative rings [RMJM 1996]; a refinement of Cohn's theorem to "nearly cancellative" monoids due to Y. Fan et al. [JA 2018]; and a characterization theorem of A. A. Antoniou and the author about atomic factorizations in certain "monoids of sets" [PJM 202?].

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Section: Applicationsmentioning

confidence: 99%

An Abstract Factorization Theorem and Some Applications

Tringali

2021

Preprint

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“…, u r 2 −1 ∈ E 1 , we have −π 1 (u r 2 ) ∈ C(π 1 (X)) by (4.3). Thus, as before, we can find a subset X 9.1) and let r 3 be the minimal index such that u…”

Section: Proofmentioning

confidence: 99%

“…The general class of semigroup treated in this paper are Transfer Krull Monoids. We remark that they include all Krull Domains, Krull Monoids, and many examples of natural semigroups of monoids, with a non-cancellative monoid of modules over Bass rings that is Transfer Krull studied in [9]. We defer the formal definitions to Section 2.3, but continue with a detailed list illustrating the broad extent of the class of semigroups covered by our results (see also [50, pp 972] and [64,Example 4.2]).…”

Section: Introductionmentioning

confidence: 99%

The Characterization of Finite Elasticities

Grynkiewicz¹

2020

Preprint

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Our main motivating goal is the study of factorization in Krull Domains H with finitely generated class group G. While factorization into irreducibles, called atoms, generally fails to be unique, there are various measures of how badly this can fail. One of the most important is the elasticity ρ(H) = lim k→∞ ρ k (H)/k, where ρ k (H) is the maximal number of atoms in any re-factorization of a product of k atoms. Having finite elasticity is a key indicator that factorization, while not unique, is not completely wild. The elasticities, as well as many other arithmetic invariants, are the same as those of an associated combinatorial monoid B(G0) of zero-sum sequences, where G0 ⊆ G are the classes containing height one primes.We characterize when finite elasticity holds for any Krull Domain with finitely generated class group G. Indeed, our results are valid for the more general class of Transfer Krull Monoids (over a subset G0 of a finitely generated abelian group G). Moreover, we show there is a minimal s ≤ (d + 1)m, where d is the torsion free rank and m is the exponent of the torsion subgroup, such that ρsOur characterization is in terms of a simple combinatorial obstruction to infinite elasticity: there existing a subset G ⋄ 0 ⊆ G0 and global bound N such that there are no nontrivial zero-sum sequences with terms from G ⋄ 0 , and every minimal zero-sum sequence has at most N terms from G0 \ G ⋄ 0 . We give an explicit description of G ⋄ 0 in terms of the Convex Geometry of G0 modulo the torsion subgroup GT ≤ G, and show that finite elasticity is equivalent to there being no positive R-linear combination of the elements of this explicitly defined subset G ⋄ 0 equal to 0 modulo GT . Additionally, we use our results to show finite elasticity implies the set of distances ∆(H), the catenary degree c(H) (for Krull Monoids) and a weakened form of the tame degree (for Krull Monoids) are all also finite, and that the Structure Theorem for Unions holds-four of the most commonly used measurements of structured factorization, after the elasticity.Our results for factorization in Transfer Krull Monoids are accomplished by developing an extensive theory in Convex Geometry generalizing positive bases. The convex cone generated byPositive bases were first introduced and studied in the mid 20th century, and the structural work initiated by Reay led to a simplified proof and strengthening of Bonnice and Klee's celebrated generalization of Carathéordory's Theorem. They have since found increasing importance in areas of applied mathematics. We show that the structural result of Reay can be extended to special types of complete simplicial fans, which we term Reay systems. We extend these results to a general theory dealing with infinite sequences of Reay systems, as well as their limit structures. The latter, while more complex, avoid the introduction of linear dependencies into the limit structure that were not originally present, circumventing the general obstacle that a limit of linearly independent sets can degenerate into linear...

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“…This is in contrast to a recent result on Bass rings (which are stable). In [ 5 , Theorem 1.1], Baeth and Smertnig show that the monoid of isomorphism classes of finitely generated torsion-free modules over a Bass ring is always transfer Krull.…”

Section: Introductionmentioning

confidence: 99%

On the arithmetic of stable domains

Bashir

Geroldinger

Reinhart

2021

Communications in Algebra

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A commutative ring R is stable if every non-zero ideal I of R is projective over its ring of endomorphisms. Motivated by a paper of Bass in the 1960s, stable rings have received wide attention in the literature ever since then. Much is known on the algebraic structure of stable rings and on the relationship of stability with other algebraic properties such as divisoriality and the 2-generator property. In the present paper, we study the arithmetic of stable integral domains, with a focus on arithmetic properties of semigroups of ideals of stable orders in Dedekind domains.

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Lattices over Bass Rings and Graph Agglomerations

Cited by 13 publications

References 47 publications

An Abstract Factorization Theorem and Some Applications

An Abstract Factorization Theorem and Some Applications

The Characterization of Finite Elasticities

On the arithmetic of stable domains

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