Inside factorial monoids and integral domains. (English) Zbl 1087.13510

Chapman, Scott T.; Halter-Koch, Franz; Krause, Ulrich

J. Algebra 252, No. 2, 350-375 (2002).

From the paper: We investigate two classes of monoids and integral domains, called inside and outside factorial, whose definitions are closely related in a divisor-theoretic manner to the concept of unique factorization.
We denote by \(\mathbb{N}\) the set of positive integers and set \(\mathbb{N}_0=\mathbb{N} \cup\{0\}\). Recall that a monoid homomorphism \(\varphi:H\to D\) is called a divisor homomorphism if, for all \(a,b\in H\), \(\varphi(a)|_D\varphi(b)\) implies \(a|_Hb\).
Definition. A monoid \(H\) is called
(a) outside factorial, if there exists a divisor homomorphism \(\varphi:H\to D\) into a factorial monoid \(D\) such that for every \(x\in D\) there exists some \(n\in\mathbb{N}\) with \(x^n\in\varphi (H)\);
(b) inside factorial, if there exists a divisor homomorphism \(\varphi: D\to H\) from a factorial monoid \(D\) such that for every \(x\in H\) there exists some \(n\in \mathbb{N}\) such that \(x^n\in\varphi(D)\).
In the paper under review, it is proved that a monoid is outside factorial if and only if it is a Krull monoid with torsion class group, and that it is inside factorial if and only if its root-closure is a rational generalized Krull monoid with torsion class group. The authors determine the structure of Cale bases of inside factorial monoids and characterize inside factorial monoids among weakly Krull monoids. These characterizations carry over to integral domains. Inside factorial orders in algebraic number fields are characterized by several other factorization properties.

MSC:

13F15	Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial)
20M14	Commutative semigroups
13G05	Integral domains
11R65	Class groups and Picard groups of orders
13A05	Divisibility and factorizations in commutative rings
11R54	Other algebras and orders, and their zeta and \(L\)-functions

Keywords:

outside factoriality; class groups; rational generalized Krull monoid; orders in algebraic number fields; monoids; integral domains; inside factorial domain

References:

[1]	(Anderson, D. D., Factorization in Integral Domains (1997), Marcel Dekker)
[2]	Chapman, S.; Geroldinger, A., Krull domains and monoids, their sets of lengths, and associated combinatorial problems, (Anderson, D. D., Factorization in Integral Domains (1997), Marcel Dekker) · Zbl 0897.13001
[3]	Fossum, R. M., The Divisor Class Group of a Krull Domain (1973), Springer · Zbl 0256.13001
[4]	Geroldinger, A.; Halter-Koch, F., On the asymptotic behaviour of lengths of factorizations, J. Pure Appl. Algebra, 77, 239-252 (1992) · Zbl 0751.20044
[5]	Geroldinger, A.; Halter-Koch, F., Arithmetical theory of monoid homomorphisms, Semigroup Forum, 48, 333-362 (1994) · Zbl 0805.20057
[6]	Halter-Koch, F., Zur Zahlen- und Idealtheorie eindimensionaler noetherscher Integritätsbereiche, J. Algebra, 136, 103-108 (1991) · Zbl 0743.13001
[7]	Halter-Koch, F., The integral closure of a finitely generated monoid and the Frobenius problem in higher dimensions, (Bonzini, C.; etal., Semigroups. Algebraic Theory and Applications to Formal Languages and Codes (1993), World Scientific), 86-93 · Zbl 0818.20079
[8]	Halter-Koch, F., Elasticity of factorizations in atomic monoids and integral domains, J. Théor. Nombres Bordeaux, 7, 367-385 (1995) · Zbl 0844.11068
[9]	Halter-Koch, F., Divisor theories with primary elements and weakly Krull domains, Boll. Un. Mat. Ital. B, 9, 417-441 (1995) · Zbl 0849.20041
[10]	Halter-Koch, F., Ideal Systems (1998), Marcel Dekker · Zbl 0953.13001
[11]	Kainrath, F., A divisor theoretic approach towards the arithmetic of Noetherian domains, Arch. Math., 73, 347-354 (1999) · Zbl 0964.13005
[12]	F. Kainrath, Elastizität endlich erzeugter Ringe, Dissertation, Graz, 1996; F. Kainrath, Elastizität endlich erzeugter Ringe, Dissertation, Graz, 1996
[13]	Krause, U., Eindeutige Faktorisierung ohne ideale Elemente, Abh. Braunschweig. Wiss. Ges., 33, 169-177 (1982) · Zbl 0518.20062
[14]	Krause, U., On monoids of finite real character, Proc. Amer. Math. Soc., 105, 546-554 (1989) · Zbl 0692.20058
[15]	Krause, U., Semigroups that are factorial from inside or from outside, (Almeida, J.; etal., Lattices, Semigroups and Universal Algebra (1990), Plenum Press) · Zbl 0736.20039
[16]	Picavet-L’Hermite, M., Factorization in some orders with a PID as integral closure, (Halter-Koch, F.; Tichy, R., Algebraic Number Theory and Diophantine Analysis (2000), de Gruyter) · Zbl 0971.13016