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1. CMB Online first
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)
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)
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 |