Arithmetic of non-principal orders in algebraic number fields

Andreas Philipp

Displaying similar documents to “Arithmetic of non-principal orders in algebraic number fields”

Elasticity of factorizations in atomic monoids and integral domains

Franz Halter-Koch (1995)

Journal de théorie des nombres de Bordeaux

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For an atomic domain $R$ , its elasticity $ρ (R)$ is defined by : $ρ (R) = sup {m / n |u_{1} \dots u_{m} = v_{1} \dots v_{n}$ for irreducible $u_{j}, v_{i} \in R}$ . We study the elasticity of one-dimensional noetherian domains by means of the more subtle invariants $μ_{m} (R)$ defined by : $μ_{m} (R) = sup {n |u_{1} \dots u_{m} = u_{1} \dots v_{n}$ for irreducible $u_{j}, v_{i} \in R}$ . As a main result we characterize all orders in algebraic number fields having finite elasticity. On the way, we obtain a series of results concerning the invariants $μ_{m}$ and $ρ$ for monoids and integral domains which are of independent interest.

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

Víctor Blanco, Pedro A. García-Sánchez, Alfred Geroldinger (2010)

Actes des rencontres du CIRM

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Arithmetical invariants—such as sets of lengths, catenary and tame degrees—describe the non-uniqueness of factorizations in atomic monoids.We study these arithmetical invariants by the monoid of relations and by presentations of the involved monoids. The abstract results will be applied to numerical monoids and to Krull monoids.

The set of minimal distances in Krull monoids

Alfred Geroldinger, Qinghai Zhong (2016)

Acta Arithmetica

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Let H be a Krull monoid with class group G. Then every nonunit a ∈ H can be written as a finite product of atoms, say $a = u_{1} \cdot . . . \cdot u_{k}$ . The set (a) of all possible factorization lengths k is called the set of lengths of a. If G is finite, then there is a constant M ∈ ℕ such that all sets of lengths are almost arithmetical multiprogressions with bound M and with difference d ∈ Δ*(H), where Δ*(H) denotes the set of minimal distances of H. We show that max Δ*(H) ≤ maxexp(G)-2,(G)-1 and that equality holds...

A structure theorem for sets of lengths

Alfred Geroldinger (1998)

Colloquium Mathematicae

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Arithmetic of block monoids

Wolfgang Alexander Schmid (2004)

Mathematica Slovaca

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Factorization properties of Krull monoids with infinite class group

Wolfgang Hassler (2002)

Colloquium Mathematicae

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For a non-unit a of an atomic monoid H we call $L_{H} (a) = k \in ℕ | a = u ₁ . . . u_{k} w i t h i r r e d u c i b l e u_{i} \in H$ the set of lengths of a. Let H be a Krull monoid with infinite divisor class group such that each divisor class is the sum of a bounded number of prime divisor classes of H. We investigate factorization properties of H and show that H has sets of lengths containing large gaps. Finally we apply this result to finitely generated algebras over perfect fields with infinite divisor class group.

The tame degree and related invariants of non-unique factorizations

Franz Halter-Koch (2008)

Acta Mathematica Universitatis Ostraviensis

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Local tameness and the finiteness of the catenary degree are two crucial finiteness conditions in the theory of non-unique factorizations in monoids and integral domains. In this note, we refine the notion of local tameness and relate the resulting invariants with the usual tame degree and the $ω$ -invariant. Finally we present a simple monoid which fails to be locally tame and yet has nice factorization properties.