Dedekind Domains
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Positivstellensatz for Semi-Algebraic Sets in Real Closed Valued Fields
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 143, Number 10, October 2015, Pages 4479–4484 http://dx.doi.org/10.1090/proc/12595 Article electronically published on April 29, 2015 POSITIVSTELLENSATZ FOR SEMI-ALGEBRAIC SETS IN REAL CLOSED VALUED FIELDS NOA LAVI (Communicated by Mirna Dˇzamonja) Abstract. The purpose of this paper is to give a characterization for polyno- mials and rational functions which admit only non-negative values on definable sets in real closed valued fields. That is, generalizing the relative Positivstellen- satz for sets defined also by valuation terms. For this, we use model theoretic tools, together with existence of canonical valuations. 1. Introduction A “Nichtnegativstellensatz” in real algebraic geometry is a theorem which gives an algebraic characterization of those polynomials admitting only non-negative val- ues on a given set. The original Nichtnegativstellensatz is the solution by Artin in [1], Theorem 45, to Hilbert’s seventeenth problem: a polynomial taking only non-negative values can be written as a sum of squares of rational functions. Later on A. Robinson [8] generalized the theorem to any real closed field using the model completeness of real closed fields [9], a model theoretic property which means that any formula is equivalent to an existential formula in the language of rings. For proof see [6, Theorem 5.1, p. 49, and Theorem 5.7, p. 54]. In real algebra, valua- tions come very naturally into the picture. A real closed field K admits a canonical valuation which is non-trivial if and only if the field is non-archimedeam, that is, not embeddable into R. -
Dedekind Domains
Dedekind Domains Mathematics 601 In this note we prove several facts about Dedekind domains that we will use in the course of proving the Riemann-Roch theorem. The main theorem shows that if K=F is a finite extension and A is a Dedekind domain with quotient field F , then the integral closure of A in K is also a Dedekind domain. As we will see in the proof, we need various results from ring theory and field theory. We first recall some basic definitions and facts. A Dedekind domain is an integral domain B for which every nonzero ideal can be written uniquely as a product of prime ideals. Perhaps the main theorem about Dedekind domains is that a domain B is a Dedekind domain if and only if B is Noetherian, integrally closed, and dim(B) = 1. Without fully defining dimension, to say that a ring has dimension 1 says nothing more than nonzero prime ideals are maximal. Moreover, a Noetherian ring B is a Dedekind domain if and only if BM is a discrete valuation ring for every maximal ideal M of B. In particular, a Dedekind domain that is a local ring is a discrete valuation ring, and vice-versa. We start by mentioning two examples of Dedekind domains. Example 1. The ring of integers Z is a Dedekind domain. In fact, any principal ideal domain is a Dedekind domain since a principal ideal domain is Noetherian integrally closed, and nonzero prime ideals are maximal. Alternatively, it is easy to prove that in a principal ideal domain, every nonzero ideal factors uniquely into prime ideals. -
8 Complete Fields and Valuation Rings
18.785 Number theory I Fall 2016 Lecture #8 10/04/2016 8 Complete fields and valuation rings In order to make further progress in our investigation of finite extensions L=K of the fraction field K of a Dedekind domain A, and in particular, to determine the primes p of K that ramify in L, we introduce a new tool that will allows us to \localize" fields. We have already seen how useful it can be to localize the ring A at a prime ideal p. This process transforms A into a discrete valuation ring Ap; the DVR Ap is a principal ideal domain and has exactly one nonzero prime ideal, which makes it much easier to study than A. By Proposition 2.7, the localizations of A at its prime ideals p collectively determine the ring A. Localizing A does not change its fraction field K. However, there is an operation we can perform on K that is analogous to localizing A: we can construct the completion of K with respect to one of its absolute values. When K is a global field, this process yields a local field (a term we will define in the next lecture), and we can recover essentially everything we might want to know about K by studying its completions. At first glance taking completions might seem to make things more complicated, but as with localization, it actually simplifies matters considerably. For those who have not seen this construction before, we briefly review some background material on completions, topological rings, and inverse limits. -
The Structure Theory of Complete Local Rings
The structure theory of complete local rings Introduction In the study of commutative Noetherian rings, localization at a prime followed by com- pletion at the resulting maximal ideal is a way of life. Many problems, even some that seem \global," can be attacked by first reducing to the local case and then to the complete case. Complete local rings turn out to have extremely good behavior in many respects. A key ingredient in this type of reduction is that when R is local, Rb is local and faithfully flat over R. We shall study the structure of complete local rings. A complete local ring that contains a field always contains a field that maps onto its residue class field: thus, if (R; m; K) contains a field, it contains a field K0 such that the composite map K0 ⊆ R R=m = K is an isomorphism. Then R = K0 ⊕K0 m, and we may identify K with K0. Such a field K0 is called a coefficient field for R. The choice of a coefficient field K0 is not unique in general, although in positive prime characteristic p it is unique if K is perfect, which is a bit surprising. The existence of a coefficient field is a rather hard theorem. Once it is known, one can show that every complete local ring that contains a field is a homomorphic image of a formal power series ring over a field. It is also a module-finite extension of a formal power series ring over a field. This situation is analogous to what is true for finitely generated algebras over a field, where one can make the same statements using polynomial rings instead of formal power series rings. -
Formal Power Series - Wikipedia, the Free Encyclopedia
Formal power series - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Formal_power_series Formal power series From Wikipedia, the free encyclopedia In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates. This perspective contrasts with that of power series, whose variables designate numerical values, and which series therefore only have a definite value if convergence can be established. Formal power series are often used merely to represent the whole collection of their coefficients. In combinatorics, they provide representations of numerical sequences and of multisets, and for instance allow giving concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved; this is known as the method of generating functions. Contents 1 Introduction 2 The ring of formal power series 2.1 Definition of the formal power series ring 2.1.1 Ring structure 2.1.2 Topological structure 2.1.3 Alternative topologies 2.2 Universal property 3 Operations on formal power series 3.1 Multiplying series 3.2 Power series raised to powers 3.3 Inverting series 3.4 Dividing series 3.5 Extracting coefficients 3.6 Composition of series 3.6.1 Example 3.7 Composition inverse 3.8 Formal differentiation of series 4 Properties 4.1 Algebraic properties of the formal power series ring 4.2 Topological properties of the formal power series -
MAXIMAL and NON-MAXIMAL ORDERS 1. Introduction Let K Be A
MAXIMAL AND NON-MAXIMAL ORDERS LUKE WOLCOTT Abstract. In this paper we compare and contrast various prop- erties of maximal and non-maximal orders in the ring of integers of a number field. 1. Introduction Let K be a number field, and let OK be its ring of integers. An order in K is a subring R ⊆ OK such that OK /R is finite as a quotient of abelian groups. From this definition, it’s clear that there is a unique maximal order, namely OK . There are many properties that all orders in K share, but there are also differences. The main difference between a non-maximal order and the maximal order is that OK is integrally closed, but every non-maximal order is not. The result is that many of the nice properties of Dedekind domains do not hold in arbitrary orders. First we describe properties common to both maximal and non-maximal orders. In doing so, some useful results and constructions given in class for OK are generalized to arbitrary orders. Then we de- scribe a few important differences between maximal and non-maximal orders, and give proofs or counterexamples. Several proofs covered in lecture or the text will not be reproduced here. Throughout, all rings are commutative with unity. 2. Common Properties Proposition 2.1. The ring of integers OK of a number field K is a Noetherian ring, a finitely generated Z-algebra, and a free abelian group of rank n = [K : Q]. Proof: In class we showed that OK is a free abelian group of rank n = [K : Q], and is a ring. -
Unique Factorization of Ideals in a Dedekind Domain
UNIQUE FACTORIZATION OF IDEALS IN A DEDEKIND DOMAIN XINYU LIU Abstract. In abstract algebra, a Dedekind domain is a Noetherian, inte- grally closed integral domain of Krull dimension 1. Parallel to the unique factorization of integers in Z, the ideals in a Dedekind domain can also be written uniquely as the product of prime ideals. Starting from the definitions of groups and rings, we introduce some basic theory in commutative algebra and present a proof for this theorem via Discrete Valuation Ring. First, we prove some intermediate results about the localization of a ring at its maximal ideals. Next, by the fact that the localization of a Dedekind domain at its maximal ideal is a Discrete Valuation Ring, we provide a simple proof for our main theorem. Contents 1. Basic Definitions of Rings 1 2. Localization 4 3. Integral Extension, Discrete Valuation Ring and Dedekind Domain 5 4. Unique Factorization of Ideals in a Dedekind Domain 7 Acknowledgements 12 References 12 1. Basic Definitions of Rings We start with some basic definitions of a ring, which is one of the most important structures in algebra. Definition 1.1. A ring R is a set along with two operations + and · (namely addition and multiplication) such that the following three properties are satisfied: (1) (R; +) is an abelian group. (2) (R; ·) is a monoid. (3) The distributive law: for all a; b; c 2 R, we have (a + b) · c = a · c + b · c, and a · (b + c) = a · b + a · c: Specifically, we use 0 to denote the identity element of the abelian group (R; +) and 1 to denote the identity element of the monoid (R; ·). -
6. PID and UFD Let R Be a Commutative Ring. Recall That a Non-Unit X ∈ R Is Called Irreducible If X Cannot Be Written As A
6. PID and UFD Let R be a commutative ring. Recall that a non-unit x R is called irreducible if x cannot be written as a product of two non-unit elements of R i.e.∈x = ab implies either a is an unit or b is an unit. Also recall that a domain R is called a principal ideal domain or a PID if every ideal in R can be generated by one element, i.e. is principal. 6.1. Lemma. (a) Let R be a commutative domain. Then prime elements in R are irreducible. (b) Let R be a PID. Then an irreducible in R is a prime element. Proof. (a) Let (p) be a prime ideal in R. If possible suppose p = uv.Thenuv (p), so either u (p)orv (p), if u (p), then u = cp,socv = 1, that is v is an unit. Similarly,∈ if v (p), then∈ u is an∈ unit. ∈ ∈(b) Let p R be irreducible. Suppose ab (p). Since R is a PID, the ideal (a, p)hasa generator, say∈ x, that is, (x)=(a, p). Then ∈p (x), so p = xu for some u R. Since p is irreducible, either u or x must be an unit and we∈ consider these two cases seperately:∈ In the first case, when u is an unit, then x = u−1p,soa (x) (p), that is, p divides a.Inthe second case, when x is a unit, then (a, p)=(1).So(∈ ab,⊆ pb)=(b). But (ab, pb) (p). So (b) (p), that is p divides b. -
Factorization in the Self-Idealization of a Pid 3
FACTORIZATION IN THE SELF-IDEALIZATION OF A PID GYU WHAN CHANG AND DANIEL SMERTNIG a b Abstract. Let D be a principal ideal domain and R(D) = { | 0 a a, b ∈ D} be its self-idealization. It is known that R(D) is a commutative noetherian ring with identity, and hence R(D) is atomic (i.e., every nonzero nonunit can be written as a finite product of irreducible elements). In this paper, we completely characterize the irreducible elements of R(D). We then use this result to show how to factorize each nonzero nonunit of R(D) into irreducible elements. We show that every irreducible element of R(D) is a primary element, and we determine the system of sets of lengths of R(D). 1. Introduction Let R be a commutative noetherian ring. Then R is atomic, which means that every nonzero nonunit element of R can be written as a finite product of atoms (irreducible elements) of R. The study of non-unique factorizations has found a lot of attention. Indeed this area has developed into a flourishing branch of Commutative Algebra (see some surveys and books [3, 6, 8, 5]). However, the focus so far was almost entirely on commutative integral domains, and only first steps were done to study factorization properties in rings with zero-divisors (see [2, 7]). In the present note we study factorizations in a subring of a matrix ring over a principal ideal domain, which will turn out to be a commutative noetherian ring with zero-divisors. To begin with, we fix our notation and terminology. -
MATH 404: ARITHMETIC 1. Divisibility and Factorization After Learning To
MATH 404: ARITHMETIC S. PAUL SMITH 1. Divisibility and Factorization After learning to count, add, and subtract, children learn to multiply and divide. The notion of division makes sense in any ring and much of the initial impetus for the development of abstract algebra arose from problems of division√ and factoriza- tion, especially in rings closely related to the integers such as Z[ d]. In the next several chapters we will follow this historical development by studying arithmetic in these and similar rings. √ To see that there are some serious issues to be addressed notice that in Z[ −5] the number 6 factorizes as a product of irreducibles in two different ways, namely √ √ 6 = 2.3 = (1 + −5)(1 − −5). 1 It isn’t hard to show these factors are irreducible but we postpone√ that for now . The key point for the moment is to note that this implies that Z[ −5] is not a unique factorization domain. This is in stark contrast to the integers. Since our teenage years we have taken for granted the fact that every integer can be written as a product of primes, which in Z are the same things as irreducibles, in a unique way. To emphasize the fact that there are questions to be understood, let us mention that Z[i] is a UFD. Even so, there is the question which primes in Z remain prime in Z[i]? This is a question with a long history: Fermat proved in ??? that a prime p in Z remains prime in Z[i] if and only if it is ≡ 3(mod 4).√ It is then natural to ask for each integer d, which primes in Z remain prime in Z[ d]? These and closely related questions lie behind much of the material we cover in the next 30 or so pages. -
Factorization in Domains
Factors Primes and Irreducibles Factorization Ascending Chain Conditions Factorization in Domains Ryan C. Daileda Trinity University Modern Algebra II Daileda Factorization Factors Primes and Irreducibles Factorization Ascending Chain Conditions Divisibility and Factors Recall: Given a domain D and a; b 2 D, ajb , (9c 2 D)(b = ac); and we say a divides b or a is a factor of b. In terms of principal ideals: ajb , (b) ⊂ (a) u 2 D× , (u) = D , (8a 2 D)(uja) ajb and bja , (a) = (b) , (9u 2 D×)(a = bu) | {z } a;b are associates Daileda Factorization Factors Primes and Irreducibles Factorization Ascending Chain Conditions Examples 1 7j35 in Z since 35 = 7 · 5. 2 2 + 3i divides 13 in Z[i] since 13 = (2 + 3i)(2 − 3i). 3 If F is a field, a 2 F and f (X ) 2 F [X ] then f (a) = 0 , X − a divides f (X ) in F [X ]: p p 4 In Z[ −5], 2 + −5 is a factor of 9: p p 9 = (2 + −5)(2 − −5): Daileda Factorization Factors Primes and Irreducibles Factorization Ascending Chain Conditions Prime and Irreducible Elements Let D be a domain and D_ = D n (D× [ f0g). Motivated by arithmetic in Z, we make the following definitions. a 2 D_ is irreducible (or atomic) if a = bc with b; c 2 D implies b 2 D× or c 2 D×; a 2 D_ is prime if ajbc in D implies ajb or ajc. We will prefer \irreducible," but \atomic" is also common in the literature. Daileda Factorization Factors Primes and Irreducibles Factorization Ascending Chain Conditions Remarks 1 p 2 Z is \prime" in the traditional sense if and only if it is irreducible in the ring-theoretic sense. -
13. Dedekind Domains 117
13. Dedekind Domains 117 13. Dedekind Domains In the last chapter we have mainly studied 1-dimensional regular local rings, i. e. geometrically the local properties of smooth points on curves. We now want to patch these local results together to obtain global statements about 1-dimensional rings (resp. curves) that are “locally regular”. The corresponding notion is that of a Dedekind domain. Definition 13.1 (Dedekind domains). An integral domain R is called Dedekind domain if it is Noetherian of dimension 1, and for all maximal ideals P E R the localization RP is a regular local ring. Remark 13.2 (Equivalent conditions for Dedekind domains). As a Dedekind domain R is an integral domain of dimension 1, its prime ideals are exactly the zero ideal and all maximal ideals. So every localization RP for a maximal ideal P is a 1-dimensional local ring. As these localizations are also Noetherian by Exercise 7.23, we can replace the requirement in Definition 13.1 that the local rings RP are regular by any of the equivalent conditions in Proposition 12.14. For example, a Dedekind domain is the same as a 1-dimensional Noetherian domain such that all localizations at maximal ideals are discrete valuation rings. This works particularly well for the normality condition as this is a local property and can thus be transferred to the whole ring: Lemma 13.3. A 1-dimensional Noetherian domain is a Dedekind domain if and only if it is normal. Proof. By Remark 13.2 and Proposition 12.14, a 1-dimensional Noetherian domain R is a Dedekind domain if and only if all localizations RP at a maximal ideal P are normal.