Related papers: Bounded and finite factorization domains

Let $A$ be a matrix with nonnegative real entries. A nonnegative factorization of size $k$ is a representation of $A$ as a sum of $k$ nonnegative rank-one matrices. The space of all such factorizations is a bounded semialgebraic set, and we…

An integral domain is said to have the IDF property when every non-zero element of it has only a finite number of non-associate irreducible divisors. A counterexample has already been found showing that IDF property does not necessarily…

We combine the language of monoids with the language of preorders so as to refine some fundamental aspects of the classical theory of factorization and prove an abstract factorization theorem with a variety of applications. In particular,…

If every subring of an integral domain is atomic, then we say that the latter is hereditarily atomic. In this paper, we study hereditarily atomic domains. First, we characterize when certain direct limits of Dedekind domains are Dedekind…

In this article we study two classes of integral domains. The first is characterized by having a finite intersection of principal ideals being finitely generated only when it is principal. The second class consists of the integral domains…

$\DeclareMathOperator{\IntR}{Int{}^\text{R}}$Integer-valued rational functions are a natural generalization of integer-valued polynomials. Given a domain $D$, the collection of all integer-valued rational functions over $D$ forms a ring…

In this paper we investigate the factorization behaviour of the binomial polynomials $\binom{x}{n} = \frac{x(x-1)\cdots (x-n+1)}{n!}$ and their powers in the ring of integer-valued polynomials $\operatorname{Int}(\mathbb{Z})$. While it is…

Let $M$ be a cancellative commutative monoid and call a submonoid $S$ of $M$ an undermonoid if $\G(S)=\G(M)$ inside the Grothendieck group of $M$. Gotti and Li asked whether the finite factorization property is hereditary once it is known…

Much work has been done on generalized factorization techniques in integral domains, namely $\tau$-factorization. There has also been substantial progress made in investigating factorization in commutative rings with zero-divisors. This…

For a positive real number $\alpha$, let $\mathbb{N}_0[\alpha,\alpha^{-1}]$ be the semiring of all real numbers $f(\alpha)$ for $f(x)$ lying in $\mathbb{N}_0[x,x^{-1}]$, which is the semiring of all Laurent polynomials over the set of…

We describe a simple approach to factorize non-commutative (nc) polynomials, that is, elements in free associative algebras (over a commutative field), into atoms (irreducible elements) based on (a special form of) their minimal linear…

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…

In 2011, Anderson and Frazier define the concept of $\tau_{(n)}$-factorization, where $\tau_{(n)}$ is a restriction of the modulo $n$ equivalence relation. These relations have been worked mostly for small values of $n$. However, it is…

A higher order difference equation may be generally defined in an arbitrary nonempty set S as: \[ f_{n}(x_{n},x_{n-1},...,x_{n-k})=g_{n}(x_{n},x_{n-1},...,x_{n-k}) \] where $f_{n},g_{n} :S^{k+1}\rightarrow S$ are given functions for…

A molecule is a nonzero non-unit element of an integral domain (resp., commutative cancellative monoid) having a unique factorization into irreducibles (resp., atoms). Here we study the molecules of Puiseux monoids as well as the molecules…

We investigate non-unique factorization of polynomials in Z_{p^n}[x] into irreducibles. As a Noetherian ring whose zero-divisors are contained in the Jacobson radical, Z_{p^n}[x] is atomic. We reduce the question of factoring arbitrary…

This paper establishes that every positive-definite matrix can be written as a positive linear combination of outer products of integer-valued vectors whose entries are bounded by the geometric mean of the condition number and the dimension…

We extend the theory of atomized semilattices to the infinite setting. We show that it is well-defined and that every semilattice is atomizable. We also study atom redundancy, focusing on complete and finitely generated semilattices and…

The notion of maximal non valuative domain is introduced and characterized. An integral domain R is called a maximal non valuative domain if R is not a valuative domain but every proper overring of R is a valuative domain. Maximal non…

In this paper, we continue the program initiated by I. Beck's now classical paper concerning zero-divisor graphs of commutative rings. After the success of much research regarding zero-divisor graphs, many authors have turned their…