Factorization Properties in Integral Domains

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 published 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 whenever 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 noetherian 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-multiplication 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}. From the deﬁnition follows that this map satisﬁes the properties:

(i) Rv = R;

∗ (ii) (aI)v = aIv, for all a ∈ K and I ∈ F(R);

(iii) I ⊆ Iv, for each I ∈ F(R);

(iv) if I ⊂ J then Iv ⊂ Jv where I,J ∈ F(R);

(v) (Iv)v = Iv for each I ∈ F(R).

A fractional ideal I ∈ F(R) is called a v−ideal or a divisorial ideal if I = Iv and the set of divisorial ideals of R is denoted by Fv(R).

The set Fv(R) is a semigroup (with unity R) with respect to the v−multiplication deﬁned by

(I,J) 7→ (IJ)v for all (I,J) ∈ Fv(R) × Fv(R).

In a Krull domain the semigroup Fv(R) becomes a group and so we deﬁne the class

Fv(R) group, denoted by C(R), as the quotient group P(R) where P(R) is the subgroup of principal ideals.

6 In Chapter 2 we focus on atomic domains and domains that satisfy accp property.

Deﬁnition 12. A domain R is said to be atomic if for each element r ∈ R∗ there ∗ exist a1, . . . , an ∈ R irreducibles such that r = a1 ··· an.

Deﬁnition 13. A domain R satisﬁes the ascending chain condition on principal ideals (accp) if each increasing chain of principal ideals of R

(r1) ⊂ (r2) ⊂ ... ⊂ (rn) ⊂ . . . rn ∈ R stabilizes.

First of all we describe Anne Grams’ example.

+ Example 2. Let K be a ﬁeld and T the additive submonoid of Q generated by i the family {1/2 pi}i≥0, where p0 = 3, p1 = 5, ... is the sequence of odd primes.

Pn ti Let A = K[X; T ] = { i0 aiX : ti ∈ T, t0 < t1 < . . . < tn and ai ∈ K} and S := {f ∈ A : f has a non-zero constant term}. Then

−1 R := S A = AS (1)

is an atomic domain which does not satisfy accp.

Then we introduce the D + M construction in order to produce other examples of atomic domains which have not accp. We also study atomicity and accp of D + M domains. We show the following result:

Proposition 1. Let T be an integral domain of the form K +M, where K is a subﬁeld of T and M is a nonzero maximal ideal of T . Let D be a subring of K and R := D+M. In particular we have the following diagram

R = D + M / D _ i π T = K + M / / T/M =∼ K Then:

(i) R is atomic if and only if T is atomic and D is a ﬁeld;

7 (ii) R satisfies accp if and only if T satisfies accp and D is a field.

We characterize atomic and accp domains also by using P(R) (the set of principal ideals of a domain R) and ∆(R) = K∗/U(R) where U(R) denotes the set of invertible elements of R. In these terms the diﬀerence between atomicity and accp is more evident. Before proceeding, we give two deﬁnitions.

Deﬁnition 14. Let (x), (y) be two principal ideals of R such that (x) ⊆ (y). Then (x) and (y) are adjacent if for each z ∈ R so that (x) ⊆ (z) ⊆ (y) then either (z) = (x) or (z) = (y).

Deﬁnition 15. Let (y) be a principal ideal of R. An ascending chain of principal ideals starting at (y)

(y) = (y0) ⊂ (y1) ⊂ (y2) ⊂ ... ⊂ (yn) ⊂ ... is maximal if (yi) and (yi+1) are adjacent for each 0 ≤ i ≤ n.

So we have the following propositions:

Proposition 2. Let R be an integral domain.

(i) R satisﬁes accp if and only if each chain in P(R) is ﬁnite;

(ii) R is atomic when for each x ∈ R∗ some maximal chain starting at x is ﬁnite.

Proposition 3. Let R be an integral domain.

(i) R satisﬁes accp if and only if each descending sequence of positive elements of ∆(R) is ﬁnite;

(ii) R is atomic if and only if each positive element of ∆(R) can be written as a ﬁnite product of minimal positive elements of ∆(R);

We study the behaviour of atomicity and accp properties in the passage to polynomial extensions (in one or more variables) and integral closure of a domain R. First we have that accp property transfers to the polynomial extensions.

8 Proposition 4. The following statements are equivalent for a domain R

(i) R satisﬁes accp;

(ii) R[X] satisﬁes accp.

If a domain R satisﬁes accp, then it is not true that the integral closure R also has accp. So we give an example of a domain that satisﬁes accp and which integral closure does not.

Example 3. Let Z be the ring of all algebraic integers and R := Z + XZ[X]. Then R = Z[X] is not even atomic since Z is not atomic. However, R satisﬁes accp.

The previous example also shows that atomicity does not transfer to integral closures and this happens with R[X] too. An example of an atomic domain whose polynomial extension in one variable is not atomic is given by M.Roitman in [21]. However we have the following partial result on atomic domains and their polynomial extensions in one variable.

Proposition 5. Let R be a domain. If R[X] is atomic, then R is atomic.

Finally we observe that when R[X] is atomic or when R satisﬁes accp then R has some properties that generalize the concepts of greatest common divisor (gcd) and least common multiple (lcm).

Deﬁnition 16. An integral domain R is strongly atomic if for each a, b ∈ R∗, 0 0 ∗ we can write a = r1 ··· rsa and b = r1 ··· rsb where r1, . . . , rs ∈ R (s ≥ 0) are irreducible, a0,b0 ∈ R∗ and gcd(a0, b0) = 1.

Deﬁnition 17. Let R be an integral domain. R is a weak gcd-domain if for each a,b ∈ R∗, a = a0r and b = b0r for some r,a0,b0 ∈ R∗ so that gcd(a0, b0) = 1.

Deﬁnition 18. An integral domain R is a lower terms domain (lt-domain) if for each a, b ∈ R∗, there exists r,r0 ∈ R with a/r = b/r0 and gcd(r, r0) = 1.

The following proposition shows the connections between these properties.

Proposition 6. Let R be an integral domain. Then:

9 (i) if R satisﬁes accp or R[X] is atomic, then R is strongly atomic;

(iii) R is strongly atomic if and only if R is an atomic weak gcd-domain;

(ii) if R is weak gcd-domain, then R is a lt-domain.

For strongly atomic domains we have a more completely result as we can see in the following proposition.

Proposition 7. The following statements are equivalent for an integral domain R:

(i) R is strongly atomic;

(ii) R[{Xα}] is atomic for any family {Xα} of indeterminates;

(iii) R[X,Y ] is atomic for indeterminates X and Y .

Let R be a gcd-domain. For each f ∈ R[X] we deﬁne the content of the polynomial f (notation: c(f)) as the greatest common divisor of its coeﬃcients. A polynomial g ∈ R[X] is said to be primitive if c(f) = 1. Then each polynomial n 0 f = a0 + a1X + ... + anX ∈ R[X] can be written as df where d = gcd(a0, . . . , an) and f 0 ∈ R[X] is primitive. An important result is that if R is strongly atomic or if R[X] is atomic, we can write each polynomial f of R[X] in the form df 0, with c(f 0) = 1, without the assumption of the existence of the greatest common divisor. The following diagram shows the implications among the classes of domains men- tioned.

accp domain ⇒ atomic domain ⇓ strongly atomic domain ⇓ gcd-domain ⇒ weak gcd-domain ⇓ lt-domain

10 In Chapter 3 we analyze bfds.

Deﬁnition 19. An atomic domain R is a bounded factorization domain (bfd) if for each r ∈ R∗ noninvertible exists a positive integer N(r) such that whenever r = r1 ··· rn as a product of irreducible elements of R, then n ≤ N(r).

Clearly a ufd is a bfd and we show that the class of accp domains contains properly the one of bfds.

+ Example 4. Let K be a ﬁeld and T the additive submonoid of Q generated by

{1/2, 1/3, 1/5,..., 1/pj,...}, where pj is the jth prime. Then R := K[X; T ] is a domain that satisﬁes accp but it is not a bfd.

In order to provide examples of bfds, we prove the following proposition.

Proposition 8. If R is a Noetherian domain or a Krull domain, then R is a bfd.

As we have done for atomicity and accp, we show under which assumptions on D the D + M construction is a bfd.

Proposition 9. Let T be an integral domain of the form K + M, where M is a non-zero maximal ideal of T and K is a subﬁeld of T . Let D be a subring of K and R = D + M. Then R is a bfd if and only if T is a bfd and D is a ﬁeld.

Then we characterize bfds in terms of P(R) and ∆(R).

Proposition 10. Let R be an integral domain.

(i) R is a bfd if and only if for each r ∈ R∗ there is a bound on the lengths of chains in P(R) starting at r.

(ii) R is a bfd if and only if each positive element rU(R) of ∆(R) must be the product of at most a ﬁxed number (depending on r) of positive elements.

Afterwards we characterize the bfds also in terms of a length function. ∗ For any atomic integral domain R, we deﬁne the function lR : R → N ∪ {∞} by putting lR(r) = 0 if r ∈ U(R) and lR(r) := sup{n : r = r1 ··· rn with each ri ∈ R irreducible} for a noninvertible element of R∗. Then

lR(xy) ≥ lR(x) + lR(y)

11 for all x, y ∈ R and it is clear that R is a bfd if and only if lR(r) < ∞ for each element ∗ belonging to R . The function lR is called length function. The following theorem shows the one-to-one correspondence between the existence ∗ of a function from R to N with the bfd property.

Theorem 4. The following conditions are equivalent for an integral domain R:

(i) R is a bfd;

(ii) for each non-invertible r ∈ R∗ there exists a positive integer N(r) such that if

r = r1 ··· rn for some ri ∈ R \U(R), then n ≤ N(r);

∗ (iii) there exists a function l : R → N such that l(u) = 0 for each u ∈ U(R) and l(xy) ≥ l(x) + l(y) for each x, y ∈ R∗.

Using this characterization we prove that bfd property is well transferred to the polynomial extension.

Proposition 11. Let R be an integral domain. The following statements are equivalent for R:

(i) R is a bfd;

(ii) R[X] is a bfd;

(iii) R[[X]] is a bfd.

The bfd property does not transfer, in general, to localizations. To show this assumption we introduce the ring of R-valued K-polynomials, that is:

Deﬁnition 20. Let A ⊂ B ⊂ K be a pair of domains with R ⊂ A. The domain n A + XB[X] = {a0 + a1X + ... + anX ∈ B[X]: a0 ∈ A} is called the composite of A and B. The domain I(B,A) = {f(X) ∈ B[X]: f(A) ⊆ A} is called the domain of A-valued B-polynomials.

12 Clearly we have A[X] ⊆ I(B,A) ⊆ A + XB[X]. Our attention is focused on I(K,R) in particular. The following corollary provide an example of a bfd whose localization is not a bfd.

Corollary 1. Let R be an integral domain with quotient field K. Then I(K,R) satisfies accp (resp. is a bfd) if and only if R satisfies accp (resp. is a bfd).

Example 5. Let R be a Dedekind domain satisfying:

(1) charR 6= 0; (2) R/P is ﬁnite for each non-zero prime ideal P, then I(K,R) is a Prüfer domain and has Krull dimension two. By Corollary (1) I(K,R) is a bfd and hence has accp. If M is a height-two maximal ideal of I(K,R), then I(K,R)M is a two-dimensional valuation domain and hence does not satisfy accp.

The example (5) is based on the fact that RS does not satisfy accp and so it also shows that accp does not transfer to localizations. However we remark that there are positive results in terms of transfer to localizations if we consider particular extensions of R and particular multiplicatively closed subset of R.

Deﬁnition 21. An extensions R ⊂ T of domains is an inert extension if whenever xy ∈ R for nonzero elements x, y of T , then there exists u ∈ U(T ) so that xu, yu−1 ∈ R.

For inert extensions we have the following property

Lemma 2. Let R ⊂ T an inert extension, then an irreducible element r of R is either irreducible or a unit in T .

Then we deﬁne a special type of multiplicative subset of R.

Deﬁnition 22. A multiplicatively closed subset is a splitting multiplicative set if for each r ∈ R there exist a ∈ R and s ∈ S such that aR ∩ tR = atR for all t ∈ S.

13 In general, if S is a multiplicatively closed subset R ⊂ RS may not be an inert extensions. However, this extension is inert if S is a splitting multiplicative set as we see in following proposition:

Proposition 12. Let R be a domain and S be a splitting multiplicative set of R, then

R ⊂ RS is an inert extension.

The following theorem asserts that if R ⊂ RS is an inert extension then atomicity, accp and bfd properties transfer to RS.

Theorem 5. Let R be a domain and S a multiplicative subset such that R ⊂ RS is an inert extensions. If RS is atomic [resp. satisﬁes accp, a bfd], then R is atomic [resp. satisﬁes accp, a bfd].

In Chapter 4 we investigate hfds which can be considered a generalization of ufds.

Deﬁnition 23. An integral domain R is a half factorial domain if R is atomic and for each non-unit r ∈ R∗, if r can be written in two diﬀerent ways as a product of irreducible elements, say r = x1 ··· xm = y1 ··· yn, then the length of these factorizations is the same, i.e. n = m.

In an attempt to provide examples of bfds which are not hfds, we study hfd property in relation to the class group of a Krull domain. There is not in general a characterization of Krull hfds but we prove that if R is Krull and the class number of R does not exceed two then R is a hfd

Theorem 6. Let R be a Krull domain with C(R) ' Z/2Z or C(R) = 0. Then R is a hfd.

Thus, since a Krull domain R is always a bfd (8) then R[X] is a bfd (since it remains Krull), while R[X] is a hfd if and only if |C(R)| ≤ 2 as we show in the following theorem.

14 Theorem 7. Let R be a Krull domain. The ring R[X] is a hfd if and only if C(R) is either 0 or Z/2Z.

As for bfds, we give a characterization of hfds in terms of a length function.

Proposition 13. The following conditions are equivalent for a domain R:

(i) R is an hfd;

∗ (ii) lR(xy) = lR(x) + lR(y) for each x, y ∈ R ;

∗ (iii) There exists a function l : R → N such that l(x) = 1 ⇔ x is irreducible in R, ∗ ∗ l(R ) = N and l(xy) = l(x) + l(y) for each x, y ∈ R .

As we have done in the other chapters we characterize hfds in terms of P(R) and ∆(R) and we see when the D + M construction is hfd.

Proposition 14. Let R be an integral domain.

(i) R is a hfd if and only if for each r ∈ R∗ every ascending maximal chains in P(R) starting at r has the same length.

(ii) R is a hfd if and only if each positive element rU(R) of ∆(R) must be the product of the same number (depending on r) of positive elements.

Proposition 15. Let T be an integral domain of the form K + M, where M is a non-zero maximal ideal of T and K is a subﬁeld of T . Let D be a subring of K and R = D + M. Then R is a hfd if and only if T is a hfd and D is a ﬁeld.

By the last proposition we give an example of a non-Noetherian hfd which is not a ufd.

Example 6. Let R := Q + XR[X]. Since R is of the form D + M, R is a hfd and R is not a Noetherian domain.

Finally by Proposition (15) we show that hfd property can not be transferred to the polynomial extension.

15 Example 7. Let R := R + Y C[Y ], R is a hfd. But in R[X] = R[Y ] + Y C[Y ][X] we have (X(1 + iY ))(X(1 − iY )) = XX(1 + Y 2), two factorizations into irreducible of diﬀerent lengths.

We remark however that there are positive results in terms of transfer to polynomial extensions of hfd property with the assumption of some additional hypothesis on the domain. Thus we prove that when R[X] is a hfd, then R is an integrally closed hfd.

Theorem 8. Let R be an integral domain. If R[X] is a hfd then R is integrally closed.

Then, by Theorem (8) we characterize Noetherian hfds whose ring of polynomials (in one or more variables) is a hfd.

Corollary 2. Let R be a Noetherian domain. Then the following statements are equivalent:

(i) R is a Krull domain with |C(R)| ≤ 2;

(ii) R[X] is a hfd;

(iii) R[X1,...,Xn] is a hfd for all n ≥ 1.

Finally, we give another example of a hfd whose polynomial extension in one variable is not a hfd. √ √ 1+ −3 Example 8. Let R = Z[ −3] and D = Z[ω−3] = Z[ 2 ]. Since R is not integrally closed we conclude that R[X] is not a hfd by Theorem (8).

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