Elasticity of factorizations in integral domains. (English) Zbl 0773.13003

Anderson, D. D.; Anderson, David F.

J. Pure Appl. Algebra 80, No. 3, 217-235 (1992).

Let \(R\) be an integral domain with quotient field \(K\). If \(R\) is a UFD, then any two factorization of a nonzero nonunit of \(R\) into the product of irreducible elements have the same length. Of course, this is not true for an arbitrary atomic integral domain (an integral domain \(R\) is atomic if each nonzero nonunit is the product of irreducible elements (atoms)). In the language of Zariski-Samuel an atomic domain is a domain satisfying the condition UF1: each nonzero nonunit is the product of irreducible elements. Following A. Zaks [Bull. Amer. Math. Soc. 82, 721-723 (1976; Zbl 0338.13020)] we define \(R\) to be a half-factorial domain (HFD) if \(R\) is atomic and whenever \(x_ 1\cdots x_ m=y_ 1\cdots y_ n\) with each \(x_ i,y_ i\in R\) irreducible, then \(m=n\). A UFD is obviously a HFD, but the converse fails since any Krull domain \(R\) with divisor class group \(\text{Cl}(R)=\mathbb{Z}_ 2\) is a HFD, but not a UFD. In order to measure how far an atomic domain \(R\) is from being a HFD, we define \(\rho(R)=\sup\{m/n| x_ 1\cdots x_ m=y_ 1\cdots y_ n\), each \(x_ i,y_ j\in R\) is irreducible}. Thus \(1\leq\rho(R)\leq\infty\), and \(\rho(R)=1\) if and only if \(R\) is a HFD. \(\rho(R)\) is called the elasticicty of \(R\) and was introduced by R. J. Valenza, who studied \(\rho(R)\) for \(R\) the ring of integers in an algebraic number field. In particular, he showed that \(\rho(R)\leq\max\{h/2,1\}\), where \(R\) has class number \(h\). In an earlier appearing (but later submitted) paper, J.- L. Steffan studied \(\rho(R)\) (without this notation) for a Dedekind domain \(R\) with finite divisor class group and showed that \(\rho(R)\leq\max\{|\text{Cl}(R)|/2,1\}\) [cf. J. Algebra 102, 229- 236 (1986; Zbl 0593.13015)].
The purpose of this paper is to study \(\rho(R)\) for an arbitrary atomic domain \(R\), but with emphasis on Krull domains. The impetus for much of this study of factorization properties goes back to the study of factorization in rings of algebraic integers, in particular, to the result of L. Carlitz that the ring of integers in an algebraic number field is a HFD if and only if it has a class number \(\leq 2\).

Reviewer: E.Stagnaro (Padova)

MSC:

13F15	Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial)
13G05	Integral domains
13C20	Class groups
11R29	Class numbers, class groups, discriminants

Keywords:

atomic integral domain; half-factorial domain; elasticicty; Krull domain

Citations:

Zbl 0338.13020; Zbl 0593.13015

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