Cale monoids, Cale domains, and Cale varieties. (English) Zbl 1100.20040

Chapman, Scott T.; Krause, Ulrich

Chapman, Scott T. (ed.), Arithmetical properties of commutative rings and monoids. Boca Raton, FL: Chapman & Hall/CRC (ISBN 0-8247-2327-9/pbk). Lecture Notes in Pure and Applied Mathematics 241, 142-171 (2005).

In some cases, two different factorizations of an element become essentially the same when raised to a power. This mostly expository paper analyzes this phenomenon in a wide range of contexts from algebraic number rings to varieties and their defining semigroup rings.
Formally, a multiplicative commutative cancellative monoid \(M\) with group of units \(M^\times\) is inside factorial with base \(Q\subset M-M^\times\) if for each \(x\in M-M^\times\), there is a positive integer \(n(x)\) and nonnegative integers \(\{t(q)\}_{q\in Q}\) such that (i) \(x^{n(x)}=u\prod_{q\in Q}q^{t(q)}\), where \(u\in M^\times\) and almost all \(t(q)\)’s are zero, and (ii) if \(x^{n(x)}=u\prod_{q\in Q}q^{t(q)}=v\prod_{q\in Q}q^{s(q)}\), where \(v\in M^\times\) and each \(s(q)\) is a nonnegative integer, then \(u=v\) and \(t(q)=s(q)\) for each \(q\in Q\). Let \(m(x)\) denote the minimal value of \(n(x)\) from (i) and \(x(q)\) the uniquely determined \(t(q)\). A base \(Q\) is a Cale base if for each \(q\in Q\), there is a positive integer \(f(q)\) such that \(f(q)x(q)/m(x)\) is an integer for all \(x\in M-M^\times\).
A Cale monoid is an inside factorial monoid with a Cale base, and a Cale domain is an integral domain \(D\) such that \(M=D-\{0\}\) is a Cale monoid. For example, \(\mathbb{Z}[\sqrt{-5}]\) and \(\mathbb{Z}_2[X^2,X^3]\) are (non UFD) Cale domains, while \(\mathbb{Z}[5\sqrt{-1}]\) and \(\mathbb{C}[X^2,X^3]\) are not.
For the entire collection see [Zbl 1061.13001].

Reviewer: David F. Anderson (Knoxville)

MSC:

20M14	Commutative semigroups
13F15	Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial)
13F05	Dedekind, Prüfer, Krull and Mori rings and their generalizations
13G05	Integral domains
13A05	Divisibility and factorizations in commutative rings
14M05	Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)

Keywords:

Cale monoids; Cale domains; Cale varieties; inside factorial commutative cancellative monoids; Cale bases; Krull monoids; groups of units