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Victor Fadinger

Publications and source records attributed to Victor Fadinger.

8 recordsLinked to original sources
arXiv2024

Primes and absolutely or non-absolutely irreducible elements in atomic domains

We give examples of atomic integral domains satisfying each of the eight logically possible combinations of existence or non-existence of the following kinds of elements: 1) primes, 2) absolutely irreducible elements that are not prime, and 3) irreducible elements that are not absolutely irreducible. A non-zero non-unit is called absolutely irreducible (or, a strong atom) if every one of its powers factors uniquely into irreducibles.

math.AC↗
arXiv2023

Lengths of factorizations of integer-valued polynomials on Krull domains with prime elements

Let $D$ be a Krull domain admitting a prime element with finite residue field and let $K$ be its quotient field. We show that for all positive integers $k$ and $1 < n_1 \leq \ldots \leq n_k$ there exists an integer-valued polynomial on $D$, that is, an element of $\text{Int}(D) = \{ f \in K[X] \mid f(D) \subseteq D \}$, which has precisely $k$ essentially different factorizations into irreducible elements of $\text{Int}(D)$ whose lengths are exactly $n_1,\ldots,n_k$. Using this, we characterize lengths of factorizations when $D$ is a unique factorization domain and therefore also in case $D$ is a discrete valuation domain. This solves an open problem proposed by Cahen, Fontana, Frisch and Glaz.

math.AC↗
arXiv2022

Integer-valued polynomials on discrete valuation rings of global fields with prescribed lengths of factorizations

Let $V$ be a valuation ring of a global field $K$. We show that for all positive integers $k$ and $1 < n_1 \leq \ldots \leq n_k$ there exists an integer-valued polynomial on $V$, that is, an element of $\text{Int}(V) = \{ f \in K[X] \mid f(V) \subseteq V \}$, which has precisely $k$ essentially different factorizations into irreducible elements of $\text{Int}(V)$ whose lengths are exactly $n_1,\ldots,n_k$. In fact, we show more, namely that the same result holds true for every discrete valuation domain $V$ with finite residue field such that the quotient field of $V$ admits a valuation ring independent of $V$ whose maximal ideal is principal or whose residue field is finite. If the quotient field of $V$ is a purely transcendental extension of an arbitrary field, this property is satisfied. This solves an open problem proposed by Cahen, Fontana, Frisch and Glaz in these cases.

math.NT↗
arXiv2022

Semigroup rings as weakly Krull domains

Let $D$ be an integral domain and $\Gamma$ be a torsion-free commutative cancellative (additive) semigroup with identity element and quotient group $G$. In this paper, we show that if char$(D)=0$ (resp., char$(D)=p>0$), then $D[\Gamma]$ is a weakly Krull domain if and only if $D$ is a weakly Krull UMT-domain, $\Gamma$ is a weakly Krull UMT-monoid, and $G$ is of type $(0,0,0, \dots )$ (resp., type $(0,0,0, \dots )$ except $p$). Moreover, we give arithmetical applications of this result.

math.AC↗
arXiv2021

On the distribution of prime divisors in Krull monoid algebras

In the present work, we prove that every class of the divisor class group of a Krull monoid algebra contains infinitely many prime divisors. Several attempts to this result have been made in the literature so far, unfortunately with open gaps. We present a complete proof of this fact.

math.AC↗
arXiv2020

On product-one sequences over subsets of groups

Let $G$ be a group and $G_0 \subseteq G$ be a subset. A sequence over $G_0$ means a finite sequence of terms from $G_0$, where the order of elements is disregarded and the repetition of elements is allowed. A product-one sequence is a sequence whose elements can be ordered such that their product equals the identity element of the group. We study algebraic and arithmetic properties of monoids of product-one sequences over finite subsets of $G$ and over the whole group $G$, with a special emphasis on infinite dihedral groups.

math.GR↗
arXiv2020

A characterization of weakly Krull monoid algebras

Let $D$ be a domain and let $S$ be a torsion-free monoid whose quotient group satisfies the ascending chain condition on cyclic subgroups. We give a characterization of when the monoid algebra $D[S]$ is weakly Krull. As corollaries, we obtain the results on when $D[S]$ is Krull resp. generalized Krull, due to Chouinard resp. El Baghdadi and Kim. Furthermore, we deduce Chang's theorem on weakly factorial monoid algebras and we characterize the weakly Krull domains among the affine monoid algebras.

math.AC↗