Factorization In Integral Domains Research Articles

On the Graph of Divisibility of an Integral Domain

AbstractIt is well known that the factorization properties of a domain are reflected in the structure of its group of divisibility. The main theme of this paper is to introduce a topological/graph-theoretic point of view to the current understanding of factorization in integral domains. We also show that connectedness properties in the graph and topological space give rise to a generalization of atomicity.

Some Results on Factorization in Integral Domains

The M-graded domains , which are almost Schreier are classified under the assumption that the integral closure of R is a root extension of R, where M is a torsion-free, commutative, cancellative monoid. In the case that D[M] is a commutative monoid domain it is shown that if M conical and is a root extension, then D[M] is almost Schreier if and only if M and D are almost Schreier. If R=ℤ[nω] is an order in a quadratic extension field of ℚ, it is shown that the conditions; R[X] is IDPF; R[X] is inside factorial; R[X] is almost Schreier; is a root extension; and every prime divisor of n also divides the discriminant of the extension K/ℚ; are equivalent conditions.

Three frameworks for a general theory of factorization

We discuss three different frameworks for a general theory of factorization in integral domains: τ-factorization, reduced τ-factorization and Γ-factorization. Let D be an integral domain, \({D^{\sharp}}\) the non-zero, non-units of D, and τ a symmetric relation on \({D^{\sharp}}\) . For \({a\in D^{\sharp}, a=\lambda a_{1}\cdots a_{n},\lambda}\) a unit, \({a_{i}\in D^{\sharp}, n\geq1,}\) and a i τa j for i ≠ j, is called a τ-factorization of a and we say a i is a τ-factor of a. For \({a,b\in D^{\sharp}, a\mid_{\tau}b}\) if a is a τ-factor of b. Then \({a\in D^{\sharp}}\) is a τ-atom if any τ-factorization of a has n = 1 and a is a τ-prime (resp., \({\mid_{\tau}}\) -prime) if \({a\mid\lambda a_{1}\cdots a_{n}}\) (resp., \({a\mid_{\tau}\lambda a_{1}\cdots a_{n}}\)), λ a1 . . . a n a τ-factorization, implies \({a\mid a_{i}}\) (resp., \({a\mid_{\tau}a_{i}}\)) for some i. The theory of reduced τ-factorization is developed similarly, except here we restrict ourselves to reduced τ-factorizations, that is, τ-factorizations a1 . . . a n where the leading unit is omitted (or is 1). The theory of Γ-factorization is as follows. For \({a\in D^{\sharp},{\rm fact}(a)}\) (resp., tfact(a)) is the set of (resp., trivial) factorizations of a, a = λ a1 . . . a n ,λ a unit, n ≥ 1 (resp,. n = 1) and \({{\rm fact}(D)=\cup_{a\in D^{\sharp}}{\rm fact}(a),{\rm tfact}(D)= \cup_{a\in D^{\sharp}}{\rm tfact}(a)}\) . Let \({\Gamma\subseteq {\rm fact}(D)}\) and \({\Gamma(a)=\Gamma\cap {\rm fact}(a)}\) ; the set of Γ-factorizations of a. For \({a,b\in D^{\sharp}, a \mid_{\Gamma} b}\) if some \({\lambda a_{1}\cdots a_{n} \in\Gamma(b)}\) has a i = a for some i. We say a is a Γ-atom if \({\Gamma(a)\subseteq {\rm tfact}(a)}\) and that a is a Γ-prime (resp., \({\mid_{\Gamma}}\) -prime) if \({a\mid\lambda a_{1} \cdots a_{n}}\) (resp., \({a\mid_{\Gamma}\lambda a_{1} \cdots a_{n}}\)) where \({\lambda a_{1}\cdots a_{n} \in\Gamma,}\) then \({a\mid a_{i}}\) (resp., \({a\mid_{\Gamma} a_{i}}\)) for some i.

On a general theory of factorization in integral domains

On a general theory of factorization in integral domains

On Factorization in Modules

In two articles, Anderson and Valdes-Leon generalized the theory of factorization in integral domains to commutative rings with zero divisors and to modules. Here we investigate some factorization properties in modules and state a result that relates factorization properties of an R-module, M, to the factorization properties of M as an (R/Ann(M))-module. Furthermore, we will investigate when a polynomial module, M[x], has the bounded factorization property, assuming that M has this property.

Factorization in Integral Domains, IV

Let D be an integral domain such that every nonzero nonunit of D is a finite product of irreducible elements. In this article, we introduce and study several unifying concepts for the theory of nonunique factorization in D. They give a new way to measure, in some sense, how far an half-factorial domain (resp., bounded factorization domain, atomic domain) D is from being a UFD (resp., finite factorization domain, Cohen–Kaplansky domain) based on equivalence relations on the set of irreducible elements of D.

Some Examples in the Theory of Factorization in Integral Domains

Let R be a UFD, S a subring of R and P be a nonzero commom ideal of R and S such that (i) R/P is a torsion-free S/P-module, (ii) if x,y ∈ S\\P and c ∈ R with y = cx then c ∈ S. In this note we study the factorization properties of such subrings and thereby obtain some examples.

Irreducible Divisor Graphs

In Beck (1988), the author introduces the idea of a zero-divisor graph of a commutative ring. We generalize this idea to study factorization in integral domains and define irreducible divisor graphs. We use these irreducible divisor graphs to characterize certain classes of domains, including UFDs.

Quadratic polynomials, factorization in integral domains and Schreier domains from pullbacks

Quadratic polynomials, factorization in integral domains and Schreier domains from pullbacks

Factorization in integral domains, III

Let A ⊂ B be an extension of integral domains and X an indeterminate. We study various factorization properties which are weaker than unique factorization in integral domains of the form A + XB[ X] or A + XB[[ X]], and in integral domains arising as pullbacks.

A Simple Example of Non-unique Factorization in Integral Domains

A Simple Example of Non-unique Factorization in Integral Domains

A Simple Example of Non-Unique Factorization in Integral Domains

(1992). A Simple Example of Non-unique Factorization in Integral Domains. The American Mathematical Monthly: Vol. 99, No. 10, pp. 943-945.

Factorization in integral domains, II

In this paper, we study several factorization properties in an integral domain which are weaker than unique factorization. We study how these properties behave under localization and directed unions.