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Über dieses Buch

This volume provides a wide-ranging survey of, and many new results on, various important types of ideal factorization actively investigated by several authors in recent years. Examples of domains studied include (1) those with weak factorization, in which each nonzero, nondivisorial ideal can be factored as the product of its divisorial closure and a product of maximal ideals and (2) those with pseudo-Dedekind factorization, in which each nonzero, noninvertible ideal can be factored as the product of an invertible ideal with a product of pairwise comaximal prime ideals. Prüfer domains play a central role in our study, but many non-Prüfer examples are considered as well.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

In this introductory chapter we introduce several variations on factoring ideals into finite products of prime ideals. For example, a domain has radical factorization if each ideal can be factored as a finite product of radical ideals. Such domains are also known as SP-domains. A domain has weak factorization if each nonzero nondivisorial ideal can be factored as the product of its divisorial closure and a finite product of maximal ideals. If one can always have such a factorization where the maximal ideals are distinct, then the domain has strong factorization. Finally, a domain has pseudo-Dedekind factorization if each nonzero noninvertible ideal can be factored as the product of an invertible ideal and a finite product of pairwise comaximal prime ideals with at least one prime in the product. In addition, if each invertible ideal has such a factorization, then the domain has strong pseudo-Dedekind factorization.

Marco Fontana, Evan Houston, Thomas Lucas

Chapter 2. Sharpness and Trace Properties

In this chapter we collect many of the definitions, properties and results that we will need in the following chapters, often improving the “classical” statements or simplifying their proofs. In particular, we consider

h

-local domains, various sharpness and trace properties (definitions are recalled in the present chapter) and we discuss their interrelations with particular attention to the Prüfer domain case and to ideal factorization. Special care has been given to the attributions of the results and to the citations of the original references.

Marco Fontana, Evan Houston, Thomas Lucas

Chapter 3. Factoring Ideals in Almost Dedekind Domains and Generalized Dedekind Domains

We start with an overview of the rings for which every proper ideal is a product of radical ideals, rings introduced by Vaughan and Yeagy under the name of SP-rings. The integral domains with this property are called here domains with radical factorization. We give several characterizations of this type of integral domains by revisiting, completing and generalizing the work by Vaughan–Yeagy (Canad. J. Math. 30:1313–1318, 1978) and Olberding (Arithmetical properties of commutative rings and monoids, Chapman & Hall/CRC, Boca Raton, 2005). In Sect. 3.2, we study almost Dedekind domains having the property that each nonzero finitely generated ideal can be factored as a finite product of powers of ideals of a factoring family (definition given below). In the subsequent section, we provide a review of the Prüfer domains in which the divisorial ideals can be factored as a product of an invertible ideal and pairwise comaximal prime ideals, after papers by Fontana–Popescu (J. Algebra 173:44–66, 1995), Gabelli (Commutative Ring Theory, Marcel Dekker, New York, 1997) and Gabelli–Popescu (J. Pure Appl. Algebra 135:237–251, 1999). The final section is devoted to the presentation of various general constructions due to Loper–Lucas (Comm. Algebra 31:45–59, 2003) for building examples of almost Dedekind (non Dedekind) domains of various kinds (e.g., almost Dedekind domains which do not have radical factorization or which have a factoring family for finitely generated ideals or which have arbitrary sharp or dull degrees (definitions given below)).

Marco Fontana, Evan Houston, Thomas Lucas

Chapter 4. Weak, Strong and Very Strong Factorization

An integral domain is said to have weak factorization if each nonzero nondivisorial ideal can be factored as the product of its divisorial closure and a finite product of (not necessarily distinct) maximal ideals. An integral domain is said to have strong factorization if it has weak factorization and the maximal ideals of the factorization are distinct. If, in addition, the maximal ideals in the factorization of a nonzero nondivisorial ideal

I

of the domain

R

can be restricted to those maximal ideals

M

such that

IR

M

is not divisorial, we say that

R

has very strong factorization. In the present section, we study these properties with particular regard to the case of Prüfer domains or almost Dedekind domains. In the Prüfer case we provide several characterizations of domains having weak, strong or very strong factorization. We discuss the connections with h-local domains and we prove that very strong and strong factorizations are equivalent for Prüfer domains.

Marco Fontana, Evan Houston, Thomas Lucas

Chapter 5. Pseudo-Dedekind and Strong Pseudo-Dedekind Factorization

The present chapter is devoted to the study of integral domains having two other kinds of ideal factorization. An integral domain is said to have strong pseudo-Dedekind factorization if each proper ideal can be factored as the product of an invertible ideal (possibly equal to the ring) and a finite product of pairwise comaximal prime ideals with at least one prime in the product. On the other hand, an integral domain is said to have pseudo-Dedekind factorization if each nonzero noninvertible ideal can be factored as the product of an invertible ideal (which might be equal to the ring) and finitely many pairwise comaximal primes. We observe that an integral domain with pseudo-Dedekind factorization has strong factorization (Sect. 4.1) and an integrally closed domain with pseudo-Dedekind factorization is an h-local Prüfer domain. Nonintegrally closed local domains with pseudo-Dedekind factorization are fully described in terms of pullbacks of valuation domains. Several characterizations of integral domains with strong pseudo-Dedekind factorization are also given. In particular, we show that an integral domain has strong pseudo-Dedekind factorization if and only if it is an h-local generalized Dedekind domain. Finally, we investigate the ascent and descent of several types of ideal factorizations from an integral domain

R

to the Nagata ring

R

(

X

) and vice versa.

Marco Fontana, Evan Houston, Thomas Lucas

Chapter 6. Factorization and Intersections of Overrings

In the first section, we introduce the notion of an

h

-local maximal ideal as a maximal ideal

M

of a domain

R

such that

$$\Theta (M){R}_{M} = K$$

(the quotient field of

R

). The second section deals with independent pairs of overrings of a domain

R

. In the case

R

can be realized as the intersection of a pair of independent overrings, we show that

R

shares various factorization properties with these overrings. For example,

R

has weak factorization if and only if both overrings have weak factorization. The third section introduces Jaffard families and Matlis partitions. Just as domains of Dedekind type are the same as

h

-local domains, a domain

R

can be realized as an intersection of the domains of a Jaffard family if and only if its set of maximal ideals can be partitioned into a Matlis partition (definitions below). As in the second section, if

$$R ={ \bigcap \nolimits }_{\alpha \in \mathcal{A}}{S}_{\alpha }$$

where

$$\{{S{}_{\alpha }\}}_{\alpha \in \mathcal{A}}$$

is a Jaffard family, then

R

satisfies a particular factoring property if and only if each

S

α

satisfies the same factoring property. The last section is devoted to constructing examples using various Jaffard families.

Marco Fontana, Evan Houston, Thomas Lucas

Backmatter

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