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Search: MSC category 13F15 ( Rings defined by factorization properties (e.g., atomic, factorial, half-factorial) [See also 13A05, 14M05] )

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1. CMB Online first

Boynton, Jason Greene; Coykendall, Jim
On the Graph of Divisibility of an Integral Domain
It 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.

Keywords:atomic, factorization, divisibility
Categories:13F15, 13A05

2. CMB 2009 (vol 53 pp. 247)

Etingof, P.; Malcolmson, P.; Okoh, F.
Root Extensions and Factorization in Affine Domains
An integral domain R is IDPF (Irreducible Divisors of Powers Finite) if, for every non-zero element a in R, the ascending chain of non-associate irreducible divisors in R of $a^{n}$ stabilizes on a finite set as n ranges over the positive integers, while R is atomic if every non-zero element that is not a unit is a product of a finite number of irreducible elements (atoms). A ring extension S of R is a \emph{root extension} or \emph{radical extension} if for each s in S, there exists a natural number $n(s)$ with $s^{n(s)}$ in R. In this paper it is shown that the ascent and descent of the IDPF property and atomicity for the pair of integral domains $(R,S)$ is governed by the relative sizes of the unit groups $\operatorname{U}(R)$ and $\operatorname{U}(S)$ and whether S is a root extension of R. The following results are deduced from these considerations. An atomic IDPF domain containing a field of characteristic zero is completely integrally closed. An affine domain over a field of characteristic zero is IDPF if and only if it is completely integrally closed. Let R be a Noetherian domain with integral closure S. Suppose the conductor of S into R is non-zero. Then R is IDPF if and only if S is a root extension of R and $\operatorname{U}(S)/\operatorname{U}(R)$ is finite.

Categories:13F15, 14A25

3. CMB 2000 (vol 43 pp. 362)

Kim, Hwankoo
Examples of Half-Factorial Domains
In this paper, we determine some sufficient conditions for an $A + XB[X]$ domain to be an HFD. As a consequence we give new examples of HFDs of the type $A + XB[X]$.

Keywords:atomic domain, HFD
Categories:13A05, 13B30, 13F15, 13G05

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