Facoltà di Scienze Matematiche, Fisiche e Naturali Corso di Laurea in Matematica Tesi di Laurea Magistrale in Matematica Factorization properties in Integral domains Summary Candidato: Relatore: Maria Chiara Timpone Prof.ssa Francesca Tartarone Anno Accademico 2012/2013 Mathematics subject classification: 13A05, 13A18, 13F05, 13F15. Key words: atomic domain, accp, bfd, hfd. Throughout this work, R will denote an integral domain (unitary commutative ring without zero divisors) with quotient field K and R∗ will denote R \{0}. The domain R is called a unique factorization domain (ufd) if each nonzero nonunit element of R can be written as a product of irreducible elements, uniquely up to order and associates. So we have the following definition. Definition 1. A domain R is a unique factorization domain (ufd) if and only if each element r of R∗ \U(R) can be written into a product of irreducible elements of R and whenever π1π2 ··· πn = ξ1ξ2 ··· ξm for irreducible elements π1, . , πn, ξ1, . , ξm of R, then 1. n = m and 2. there exists a permutation σ ∈ Sn such that for each 1 ≤ i ≤ n, πi = uiξσ(i) for some ui ∈ U(R). We say that a domain R satisfies the ascending chain condition on principal ideals (accp) if each increasing sequence of principal ideals in R (r1) ⊂ (r2) ⊂ ... ⊂ (rn) ⊂ . rn ∈ R stabilizes. This property guarantees the existence of the factorization for any nonzero nonunit element of a domain R. Therefore ufds are always a accp domains. The purpose of this thesis is to investigate different types of factorization properties in integral domains. Specifically we study these properties in some particular cases and their behaviour in certain types of extensions of integral domains such as localizations, polynomial extensions and integral closures. It is well known that ufds have many nice properties. Among them we recall that if R is a ufd then so are its polynomial extension in one variable R[X], its localizations S−1R and its integral closure R. For this reason we try to figure out if this also happens with domains that satisfies other properties of factorization. Furthermore we give many examples and counterexamples of the implications between these properties. 1 We start with a brief historical introduction about factoriality in the last century. The concept of half-factorial domain was introduced implicitly in a paper by Leonard Carlitz 1960. The paper was written to answer to a challenge of Narkiewicz who asked for a characterization of rings of algebraic integers in terms of the class number (the cardinality of the class group). In 1960, Carlitz partial answer to this question appeared in the Proceeding of the American Mathematical Society [7]. The main result of this paper is that if R is a ring of algebraic integers then one has unique length of factorizations if and only if the class number does not exceed two, [7]. In 1968 Paul Moritz Cohn wrote a paper on Bezout rings (rings whose finitely generated ideals are principals) [8] in which he defined the atom to be a nonunit element of a ring that cannot be written as a product of two nonunits and he introduced atomic domains as domains in which each nonzero nonunit element can be written as a finite product of atoms. Six years later, Mathematical Proceedings of the Cambridge Philosophical Society Journal published a work by Anne Grams [14]. In this paper she gave an example of an atomic domain which does not satisfy the accp property. Few years later, precisely in 1978 and in 1980, Abraham Zaks published a couple of papers [22, 23] on “unique length of factorization” property and coined the terminology “half factorial domain”. He defined half factorial domain (hfd) a domain R such that whenever a noninvertible element r of R has two irreducible factorizations π1 ··· πn = ξ1 ··· ξm then n = m (see 23). So, in a certain sense, a hfd is just a ufd with “half of the axioms”. In 1990, Daniel D. Anderson, David F. Anderson and Muhammad Zafrullah pub- lished the paper Factorization in integral domains [1] in which they investigated sev- eral properties of factorization. Two years after the authors continued their study in a sequel [2]. The goal of these articles was to give a careful study of various related factorization properties weaker than unique factorization. 2 This thesis is divided into two parts. In the first part (Chapter 1) we give a brief introduction of some tools necessary in the study of properties of factorization. The second part is divided in three chapters (2, 3, 4); each of these is focused on a particular property of factorization. In particular we analyze atomicity, accp, bfd and hfd properties and the connections among these properties. In Chapter 1 we introduce the concept of divisibility and the notions of prime and irreducible element. Definition 2. An element r ∈ R is said to be prime if it is nonzero nonunit and whenever r|ab with a,b ∈ R, then r|a or r|b. Definition 3. An element r ∈ R nonzero nonunit is said to be irreducible if when- ever r = ab with a, b ∈ R, a or b must be a unit in R. The existence of these elements in a domain represents the basis for the study of factorization properties (there are domains in which these elements do not exist; see ex. (1)). In relation to the notions of prime and irreducible elements, we introduce the concepts of greatest common divisor and least common multiple. Definition 4. Let a, b ∈ R, b 6= 0. The greatest common divisor of a and b (notation: gcd(a, b)) is an element d ∈ R∗ such that (i) d|a and d|b; (ii) if there exists d0 ∈ R such that d0|a and d0|b then d0|d. A domain R in which there exists gcd of any couple of elements (not both zero) of R is called a gcd-domain. Definition 5. Let a, b ∈ R. The least common multiple of a and b (notation: lcm(a, b)) is an element m ∈ R such that (i) a|m and b|m; (ii) if there exists m0 ∈ R such that a|m0 and b|m0 then m|m0. 3 A domain R in which there exists lcm of any couple of elements is called a lcm- domain. Using gcd and lcm, we prove the following lemma. Lemma 1 (Euclid’s Lemma). Let R be a gcd-domain. If a|bc and gcd(a, b) = 1, then a|c. In particular we have that: Theorem 1. Let R be a ufd. An element p ∈ R is prime if, and only if, is irreducible. Always in Chapter 1 we briefly analyze and characterize some well known classes of domains. Theorem 2. Let R be a domain. The following conditions are equivalent: (i) R satisfies the maximal condition; (ii) Every ideal of R is finitely generated; (iii) R satisfies the ascending chain condition on ideals. By using the following characterization of integral domains, we introduce noethe- rian domains. Definition 6. A domain R satisfying one of the equivalent conditions of Theorem 2 is called a noetherian domain. A domain R is local if R has exactly one maximal ideal. An important class of local domains is given by valuation domains. Definition 7. A domain R with quotient field K is a valuation domain if for every nonzero α ∈ K, either α or α−1 belongs to R In fact we see that: Theorem 3. If R is a valuation domain, then R is a local domain. By using the theory of valuations, we give an example of a domain that does not have irreducible elements. 4 Example 1. Let V = (V, M) be a valuation domain whose maximal ideal M is not principal, then V does not have irreducible elements. Particular valuation domains are discrete valuation domains. Definition 8. A domain R is a discrete valuation domain (dvr) if it is a one- dimensional valuation domain, whose maximal ideal is principal. In particular a valuation domain is ufd if and only if it is a dvr. One of the most important generalizations of valuation domains is given by Dedekind domains. Definition 9. A Dedekind domain is an integral domain R satisfying the following three conditions: 1. R is a Noetherian domain; 2. R is integrally closed; 3. Every nonzero prime ideal of R is maximal. Equivalently a domain R is Dedekind if and only if RM is a dvr for each M ∈ Max(R). Using the theory of fractional ideals (that are subsets I of K such that I is an R-module and there exists a nonzero element d ∈ R so that dI ⊆ R), we give the notion of Prüfer domains. Definition 10. A prüfer domain is a domain R such that each nonzero finitely generated fractional ideal of R is invertible. Finally we present the main generalization of ufds: krull domain. Definition 11. Let R be an integral domain. R is a Krull domain if R can be written as intersection with finite character of dvrs, that is \ R = Rλ λ∈Λ ∗ where Rλ is a dvr ∀λ ∈ Λ, and for each element r ∈ R , r is noninvertible in at most a finite number of Rλ. 5 We give the definitions of divisorial closure, divisorial ideals and v-multi- plication and we investigate some of their properties in order to construct the class group of a Krull domain. The v−operation, or divisorial closure, is a map from the set F(R) of nonzero fractional ideals of R to itself that is defined by setting Iv := (R :(R : I)) for each I ∈ F(R) with (R : I) = {x ∈ K : xI ⊆ R}.