arXiv subjects

Qinghai Zhong

Publications and source records attributed to Qinghai Zhong.

At least 19 recordsLinked to original sources

arXiv2026

On zero-sum problems over metacyclic groups $C_n \rtimes_s C_2$

Let $G$ be a finite group. A finite collection of elements from $G$, where the order is disregarded and repetitions are allowed, is said to be a product-one sequence if its elements can be ordered such that their product in $G$ equals the identity element of $G$. Then, the Gao's constant $\mathsf E (G)$ of $G$ is the smallest integer $\ell$ such that every sequence of length at least $\ell$ has a product-one subsequence of length $|G|$. For a positive integer $n$, we denote by $C_n$ a cyclic group of order $n$. Let $G = C_n \rtimes_s C_2$ with $s^2\equiv 1\pmod n$ be a metacyclic group. The direct and inverse problems of $\mathsf E (G)$ were settled recently, except for the case that $G=C_{3n_2}\rtimes_s C_2$ with $n_2\neq 1$, $\gcd(n_2,6)=1$, $s\equiv -1 \pmod 3$, and $s\equiv 1\pmod {n_2}$. In this paper, we complete the remaining case and hence for all metacyclic groups of the form $G=C_n \rtimes C_2$, the Gao's constant and the associated inverse problem are now fully settled (see Theorem 1.2).

math.CO↗

arXiv2026

On separating sets of polynomial invariants of finite abelian group actions

Let $G$ be a finite group acting on a finite dimensional complex vector space $V$ via linear transformations. Let $\mathbb{C}[V]^G$ be the algebra of polynomials that are invariant under the induced $G$-action on the polynomial ring $\mathbb{C}[V]$. A subset $S\subseteq\mathbb{C}[V]^G$ is a separating set if it separates the orbits of the group action. If $G$ is abelian, then there exist finite separating sets consisting of monomials. In this paper we investigate properties of separating sets from four different points of view, including the monoid theoretical properties of separating sets consisting of monomials, the minimal size of separating sets consisting of monomials, the exact value of the separating Noether number $\sepbeta(G)$ of abelian groups of rank $4$, and the inverse problem of $\sepbeta(G)$ for abelian groups of rank $2$.

math.AC↗

arXiv2025

On conductor submonoids of factorial monoids

We study algebraic and arithmetic properties of submonoids (resp. subrings) of factorial monoids (resp. factorial domains) whose non-invertible elements all lie in the conductor. This continues earlier work of Baeth, Cisto, et al.. On our way we answer several conjectures, formulated in their papers in the affirmative ([1,Conjecture 4.16] and [6, Conjectures 2.3 and 2.10, and Section 9]).

math.AC↗

arXiv2025

On the inverse problem of the $k$-th Davenport constants for groups of rank $2$

For a finite abelian group $G$ and a positive integer $k$, let $\mathsf{D}_k(G)$ denote the smallest integer $\ell$ such that each sequence over $G$ of length at least $\ell$ has $k$ disjoint nontrivial zero-sum subsequences. It is known that $\mathsf D_k(G)=n_1+kn_2-1$ if $G\cong C_{n_1}\oplus C_{n_2}$ is a rank $2$ group, where $1<n_1\t n_2$. We investigate the associated inverse problem for rank $2$ groups, that is, characterizing the structure of zero-sum sequences of length $\mathsf D_k(G)$ that can not be partitioned into $k+1$ nontrivial zero-sum subsequences.

math.CO↗

arXiv2025

On the separating Noether number of finite abelian groups

The separating Noether number $β_{\mathrm{sep}}(G)$ of a finite group $G$ is the minimal positive integer $d$ such that for every finite $G$-module $V$ there is a separating set consisting of invariant polynomials of degree at most $d$. In this paper we use methods from additive combinatorics to investigate the separating Noether number for finite abelian groups. Among others, we obtain the exact value of $β_{\mathrm{sep}}(G)$, provided that $G$ is either a $p$-group or has rank $2$, $3$ or $5$.

math.AC↗

arXiv2023

On Monoids of plus-minus weighted Zero-Sum Sequences: The Isomorphism Problem and the Characterization Problem

Let $G$ be an additive abelian group. A sequence $S=g_1\cdot\ldots\cdot g_{\ell}$ of terms from $G$ is a plus-minus weighted zero-sum sequence if there are $\varepsilon_1,\ldots,\varepsilon_{\ell}\in\{-1,1\}$ such that $\varepsilon_1 g_1+\ldots+\varepsilon_{\ell} g_{\ell}=0$. We first characterize (in terms of $G$) when the monoid $\mathcal{B}_{\pm}(G)$ of plus-minus weighted zero-sum sequences is Mori resp. Krull resp. finitely generated. After that we study the Isomorphism and the Characterization Problem for monoids of plus-minus weighted zero-sum sequences.

math.AC↗

arXiv2022

On monoids of weighted zero-sum sequences and applications to norm monoids in Galois number fields and binary quadratic forms

Let $G$ be an additive finite abelian group and $Γ\subset \operatorname{End} (G)$ be a subset of the endomorphism group of $G$. A sequence $S = g_1 \cdot \ldots \cdot g_{\ell}$ over $G$ is a ($Γ$-)weighted zero-sum sequence if there are $γ_1, \ldots, γ_{\ell} \in Γ$ such that $γ_1 (g_1) + \ldots + γ_{\ell} (g_{\ell})=0$. We construct transfer homomorphisms from norm monoids (of Galois algebraic number fields with Galois group $Γ$) and from monoids of positive integers, represented by binary quadratic forms, to monoids of weighted zero-sum sequences. Then we study algebraic and arithmetic properties of monoids of weighted zero-sum sequences.

math.NT↗

arXiv2021

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↗

arXiv2021

A characterization of length-factorial Krull monoids

An atomic monoid is length-factorial if each two distinct factorizations of any element have distinct factorization lengths. We provide a characterization of length-factorial Krull monoids in terms of their class groups and the distribution of prime divisors in the classes.

math.AC↗

arXiv2021

A realization result for systems of sets of lengths

Let $\mathcal L^*$ be a family of finite subsets of $\mathbb N_0$ having the following properties. (a). $\{0\}, \{1\} \in \mathcal L^*$ and all other sets of $\mathcal L^*$ lie in $\mathbb N_{\ge 2}$. (b). If $L_1, L_2 \in \mathcal L^*$, then the sumset $L_1 + L_2 \in \mathcal L^*$. We show that there is a Dedekind domain $D$ whose system of sets of lengths equals $\mathcal L^*$.

math.AC↗

arXiv2020

On clean, weakly clean, and feebly clean commutative group rings

A ring $R$ is said to be clean if each element of $R$ can be written as the sum of a unit and an idempotent. $R$ is said to be weakly clean if each element of $R$ is either a sum or a difference of a unit and an idempotent, and $R$ is said to be feebly clean if every element $r$ can be written as $r=u+e_1-e_2$, where $u$ is a unit and $e_1,e_2$ are orthogonal idempotents. Clearly clean rings are weakly clean rings and both of them are feebly clean. In a recent article (J. Algebra Appl. 17 (2018), 1850111(5 pages)), McGoven characterized when the group ring $\mathbb Z_{(p)}[C_q]$ is weakly clean and feebly clean, where $p, q$ are distinct primes. In this paper, we consider a more general setting. Let $K$ be an algebraic number field, $\mathcal O_K$ its ring of integers, $\mathfrak p\subset \mathcal O$ a nonzero prime ideal, and $\mathcal O_{\mathfrak p}$ the localization of $\mathcal O$ at $\mathfrak p$. We investigate when the group ring $\mathcal O_{\mathfrak p}[G]$ is weakly clean and feebly clean, where $G$ is a finite abelian group, and establish an explicit characterization for such a group ring to be weakly clean and feebly clean for the case when $K=\mathbb Q(ζ_n)$ is a cyclotomic field or $K=\mathbb Q(\sqrt{d})$ is a quadratic field.

math.RA↗

arXiv2020

On product-one sequences over dihedral groups

Let $G$ be a finite group. A sequence over $G$ means a finite sequence of terms from $G$, where repetition is allowed and the order is disregarded. A product-one sequence is a sequence whose elements can be ordered such that their product equals the identity element of the group. The set of all product-one sequences over $G$ (with concatenation of sequences as the operation) is a finitely generated C-monoid. Product-one sequences over dihedral groups have a variety of extremal properties. This article provides a detailed investigation, with methods from arithmetic combinatorics, of the arithmetic of the monoid of product-one sequences over dihedral groups.

math.NT↗

arXiv2020

On a zero-sum problem arising from factorization theory

We study a zero-sum problem dealing with minimal zero-sum sequences of maximal length over finite abelian groups. A positive answer to this problem yields a structural description of sets of lengths with maximal elasticity in transfer Krull monoids over finite abelian groups.

math.CO↗

arXiv2020

On an inverse problem of Erd\H os, Kleitman, and Lemke

Let $(G, 1_G)$ be a finite group and let $S=g_1\bdot \ldots\bdot g_{\ell}$ be a nonempty sequence over $G$. We say $S$ is a tiny product-one sequence if its terms can be ordered such that their product equals $1_G$ and $\sum_{i=1}^{\ell}\frac{1}{\ord(g_i)}\le 1$. Let $\mathsf {ti}(G)$ be the smallest integer $t$ such that every sequence $S$ over $G$ with $|S|\ge t$ has a tiny product-one subsequence. The direct problem is to obtain the exact value of $\mathsf {ti}(G)$, while the inverse problem is to characterize the structure of long sequences over $G$ which have no tiny product-one subsequences. In this paper, we consider the inverse problem for cyclic groups and we also study both direct and inverse problems for dihedral groups and dicyclic groups.

math.NT↗

arXiv2019

Factorization Theory in Commutative Monoids

This is a survey on factorization theory. We discuss finitely generated monoids (including affine monoids), primary monoids (including numerical monoids), power sets with set addition, Krull monoids and their various generalizations, and the multiplicative monoids of domains (including Krull domains, rings of integer-valued polynomials, orders in algebraic number fields) and of their ideals. We offer examples for all these classes of monoids and discuss their main arithmetical finiteness properties. These describe the structure of their sets of lengths, of the unions of sets of lengths, and their catenary degrees. We also provide examples where these finiteness properties do not hold.

math.AC↗

arXiv2019

Clean group rings over localizations of rings of integers

A ring $R$ is said to be clean if each element of $R$ can be written as the sum of a unit and an idempotent. In a recent article (J. Algebra, 405 (2014), 168-178), Immormino and McGoven characterized when the group ring $\mathbb Z_{(p)}[C_n]$ is clean, where $\mathbb Z_{(p)}$ is the localization of the integers at the prime $p$. In this paper, we consider a more general setting. Let $K$ be an algebraic number field, $\mathcal O_K$ be its ring of integers, and $R$ be a localization of $\mathcal O_K$ at some prime ideal. We investigate when $R[G]$ is clean, where $G$ is a finite abelian group, and obtain a complete characterization for such a group ring to be clean for the case when $K=\mathbb Q(ζ_n)$ is a cyclotomic field or $K=\mathbb Q(\sqrt{d})$ is a quadratic field.

math.RA↗

arXiv2019

On half-factoriality of transfer Krull monoids

Let $H$ be a transfer Krull monoid over a subset $G_0$ of an abelian group $G$ with finite exponent. Then every non-unit $a\in H$ can be written as a finite product of atoms, say $a=u_1 \cdot \ldots \cdot u_k$. The set $\mathsf L(a)$ of all possible factorization lengths $k$ is called the set of lengths of $a$, and $H$ is said to be half-factorial if $|\mathsf L(a)|=1$ for all $a\in H$. We show that, if $a \in H$ and $|\mathsf L(a^{\lfloor (3\exp(G) - 3)/2 \rfloor})| = 1$, then the smallest divisor-closed submonoid of $H$ containing $a$ is half-factorial. In addition, we prove that, if $G_0$ is finite and $|\mathsf L(\prod_{g\in G_0}g^{2\mathsf{ord}(g)})|=1$, then $H$ is half-factorial.

math.AC↗

arXiv2019

On Erdős-Ginzburg-Ziv inverse theorems for Dihedral and Dicyclic groups

Let $G$ be a finite group and exp$(G)$ = lcm$\{$ord$(g)$$\mid$$g \in G \}$. A finite unordered sequence of terms from $G$, where repetition is allowed, is a product-one sequence if its terms can be ordered such that their product equals the identity element of $G$. We denote by $\mathsf s (G)$ (or $\mathsf E (G)$ respectively) the smallest integer $\ell$ such that every sequence of length at least $\ell$ has a product-one subsequence of length $\exp (G)$ (or $|G|$ respectively). In this paper, we provide the exact values of $\mathsf s (G)$ and $\mathsf E (G)$ for Dihedral and Dicyclic groups and we provide explicit characterizations of all sequences of length $\mathsf s (G) - 1$ (or $\mathsf E (G) - 1$ respectively) having no product-one subsequence of length $\exp (G)$ (or $|G|$ respectively).

math.CO↗