Related papers: On congruence half-factorial Krull monoids with cy…

In our previous work, motivated by the study of tropical polynomials, a definition for prime congruences was given for an arbitrary commutative semiring. It was shown that for additively idempotent semirings this class exhibits some…

For rings R with identity, we define a class of nonlinear higher order recurrences on unitary left R-modules that include linear recurrences as special cases. We obtain conditions under which a recurrence of order k+1 in this class is…

This paper deals with function field analogues of famous theorems of Laudau which counted the number of integers which have $t$ prime factors and R. Hall which researched the distribution of divisors of integers in residue classes.\;We…

The investigation of the ideal class group $Cl_K$ of an algebraic number field $K$ is one of the key subjects of inquiry in algebraic number theory since it encodes a lot of arithmetic information about K. There is a considerable amount of…

In this paper we study probabilistic aspects such as (cyclic) subgroup commutativity degree and (cyclic) factorization number of ZM-groups. We show that these quantities can be computed using the sizes of the conjugacy classes of these…

We introduce a new invariant describing the structure of sets of lengths in atomic monoids and domains. For an atomic monoid $H$, let $\Delta_{\rho} (H)$ be the set of all positive integers $d$ which occur as differences of arbitrarily long…

Let K/Q be a cyclic extension. In this paper, we give several congruences connecting the prime divisors of the degree g= [K:Q] with the prime divisors of the class number h of K/Q. As an exemple, the theorem: Let K/Q be a cyclic extension…

We investigate from an algebraic and topological point of view the minimal prime spectrum of a universal algebra, considering the prime congruences w.r.t. the term condition commutator. Then we use the topological structure of the minimal…

We study a numerical semigroup ring as an algebra over another numerical semigroup ring. The complete intersection property of numerical semigroup algebras is investigated using factorizations of monomials into minimal ones. The goal is to…

Let $q\geqslant 2$ be a fixed prime power. We prove an asymptotic formula for counting the number of monic polynomials that are of degree $n$ and have exactly $k$ irreducible factors over the finite field $\mathbb{F}_q$. We also compare our…

The set of distances of a monoid or of a domain is the set of all $d \in \mathbb N$ with the following property: there are irreducible elements $u_1, \ldots, u_k, v_1, \ldots, v_{k+d}$ such that $u_1 \cdot \ldots \cdot u_k = v_1 \cdot…

We develop first steps in the study of factorizations of elements in ultraproducts of commutative cancellative monoids into irreducible elements. A complete characterization of the (multi-)sets of lengths in such objects is given. As…

Congruence families, i.e., $\ell$-adic convergence for well-defined arithmetic subsequences, is a commonplace phenomenon for the coefficients of modular forms. Such families superficially resemble one another, but they often vary…

The arithmetic properties of the ordinary partition function $p(n)$ have been the topic of intensive study for the past century. Ramanujan proved that there are linear congruences of the form $p(\ell n+\beta)\equiv 0\pmod\ell$ for the…

Let $\mathbb{N} \mathcal{A}$ be the monoid generated by $\mathcal{A} = {\mathbf{a}_1, ..., \mathbf{a}_n} \subseteq \mathbb{Z}^d.$ We introduce the homogeneous catenary degree of $\mathbb{N} \mathcal{A}$ as the smallest $N \in \mathbb N$…

In Section 1 of the paper, we prove McCoy's property for the zero-divisors of polynomials in semirings. We also investigate zero-divisors of semimodules and prove that under suitable conditions, the monoid semimodule $M[G]$ has very few…

In this article we study the combinatorics of congruence subgroups of the modular group. We consider the notion of minimal monomial solutions. These are the solutions of a matrix equation (also appearing in the study of Coxeter friezes),…

Let $R$ be an order in an algebraic number field. If $R$ is a principal order, then many explicit results on its arithmetic are available. Among others, $R$ is half-factorial if and only if the class group of $R$ has at most two elements.…

We introduce the notion of independent sequences with respect to a monomial order by using the least terms of polynomials vanishing at the sequence. Our main result shows that the Krull dimension of a Noetherian ring is equal to the…

Factorial k-means (FKM) clustering is a method for clustering objects in a low-dimensional subspace. The advantage of this method is that the partition of objects and the low-dimensional subspace reflecting the cluster structure are…