Bachelorarbeit

Total Page:16

File Type:pdf, Size:1020Kb

Bachelorarbeit Bachelorarbeit Zur Erlangung des akademischen Grades Bachelor of Science Dimension theory for commutative rings Eingereicht von: Daniel Heiß Eingereicht bei: Prof. Dr. Niko Naumann Universität Regensburg Fakultät für Mathematik Ausgabetermin: 22.08.2013 (Semester 5) Abgabetermin: 25.09.2013 Contents Contents I Introduction 1 II Primary Decomposition 2 2.1 Primary ideals . .2 2.2 The Lasker-Noether decomposition theorem . .4 III Krull’s principal ideal theorem 6 3.1 The n-th symbolic powers . .6 3.2 Krull’s principal ideal theorem . .7 IV A first boundary for dim(R[X]) 11 V Dimension theory of K[X1;:::;Xn] 14 VI Dimension theory of Noetherian rings 16 6.1 Dimension under polynomial extensions . 16 6.2 Non-transcendental extensions . 17 Summary of non-transcendental extensions . 21 VII Prüfer domains and a note on Seidenberg’s F -rings 21 7.1 Dimension theory of Prüfer domains . 21 7.2 A note on F -rings . 22 Summary of polynomial extensions . 27 VIII A journey to the power series ring 27 Summary and comparison with polynomial extensions . 29 References iii ii I Introduction I Introduction In mathematics there are several concepts of dimension. Some of them are quite intuitive, like the Hamel dimension: The Hamel dimension of a vector space is the minimal number of n vectors that are needed to generate the vector space. For the euclidean space R this coin- cides with the “natural” term of dimension which is the minimal number of coordinates that are necessary to specify each point of the space. On the other hand there are less intuitive concepts of dimension: Studying fractals one usually comes across the Hausdorff dimension which associates to any metric space a certain non-negative real number, which (for fractals) can be non-integer. For example the Sierpinski triangle (a famous fractal) has the Hausdorff log(3) dimension log(2) 2= Q. In algebraic geometry one of the basic objects of study are affine algebraic varieties. The dimension of an affine algebraic variety Y is defined as the supremum of all natural numbers n 2 N such that there exists a chain Z0 ( Z1 ( ::: ( Zn of irreducible closed subsets of Y . This idea of dimension is clearly the generalization of the intuitive Hamel dimension; and it can be translated into notion of commutative algebra: For this let k be an algebraically closed field and m ≥ 1 an integer. For an affine algebraic subvariety Y of km the chain of irre- ducible closed subsets of Y correspond by Hilbert’s Nullstellensatz to a chain of prime ideals p0 ( p1 ( ::: ( pn of k[X1;:::;Xm] containing the vanishing ideal I(Y ) of Y . This chain in turn corresponds to a chain of prime ideals p¯0 ( ::: ( p¯n in the ring k[X1;:::;Xm]=I(Y ). So the dimension of a variety Y can be interpreted as the supremum of the lengths of chains of prime ideals in the ring k[X1;:::;Xm]=I(Y ). So we translated the concept of geometric dimension into notion of pure algebra. There this concept of dimension is called the Krull dimension of a ring and throughout this thesis we will call it just “the dimension”. Clearly the Krull dimension of any field k is zero and any principal ideal domain that is not a field is one-dimensional. The main interest of this thesis will be the behavior of the dimension of a commutative unitary ring R under certain extensions R −! S. Mainly we study polynomial extensions and see that in several cases the dimension rises by the number of adjoint indeterminates. This thesis is organized as follows: We start with a brief introduction to primary de- composition in order to proof Krull’s principal ideal theorem and similar results in section III. Equipped with this we are able to give a lower and an upper bound for the dimension dim(R[X]) of the polynomial ring R[X] as it is provided by [Sei53]. Then we follow princi- pally [BMRH73] and we will give an elementary proof that dim k[X1;:::;Xn] = n for any field k or more generally any Artinian ring. Next we pass from Artinian rings to Noethe- rian rings and we will see – still following [BMRH73] – that we have a similar behavior: For any Noetherian ring we have dim(R[X]) = dim(R) + 1 fitting the previous result for zero-dimensional Noetherian rings. Still in the Noetherian case and now following [Cla65] we will further study the Krull dimension under ring extensions R −! R[x] where x is not an indeterminate. It will turn out that the dimension either decreases by one or does not 1 II Primary Decomposition change at all and we will give some necessary and sufficient conditions for the dimension to decrease. In section VII we will find out that the beautiful property of the dimension under polynomial extensions in the case of Noetherian rings is also true for semi-hereditary rings, especially it is true for Prüfer domains. Further we will discuss rings that do not behave like Noetherian rings or Prüfer domains under polynomial extentions. Especially we will have a look at F -rings provided by Seidenberg in [Sei53, Sei54]. Eventually we will make a short visit to the formal power series rings in one and several indeterminates observing that in the Noetherian case the behavior is just like in the polynomial case while in the non-Noetherian case various weird things can happen. In this last section we only want to give a rough overview of Arnold’s work in [Arn73a,Arn73b,Arn82] concerning the large theory about the Krull dimension under power series extensions. Notation. Throughout the whole thesis R denotes a commutative unitary non-zero ring of finite Krull dimension. Further the set of natural numbers N is supposed to contain zero. If a ring does not contain any proper zero-divisors we call it a domain. Moreover if not otherwise specified X; X1;:::;Xn denote indeterminates and p ⊆ R is assumed to be a prime ideal. The set of all prime ideals of R is denoted by Spec(R). When passing to a residue class ring R=a for some ideal a we denote with x¯ the residue class of x for any element x 2 R and similar by b¯ we mean the image of the ideal b. II Primary Decomposition In this section we will briefly develop the theory of primary decomposition which is a basic tool studying ideals. Mainly we follow [ZS75], but also [AM69]. 2.1 Primary ideals We start with some basic facts about the radical of an ideal. p n o n Definition 2.1. Let a ⊆ R be an ideal. Then we call a := x 2 R 9n 2 N: x 2 a the radical of a. p p Lemma 2.2. (i) If a ⊆ b for some ideals a; b ⊆ R, then a ⊆ b. p (ii) For any prime ideal p 2 Spec(R) the equation pn = p holds for all n ≥ 1. (iii) Let I be a finite set and for all i 2 I let qi ⊆ R be an ideal. Then we have pT T p i2I qi = i2I qi. p (iv) Let a ⊆ R be an ideal. Then a is the intersection of all prime ideals p 2 Spec(R) containing a. Proof. Those are well-known properties. See [AM69, 1.13f.] among others. 2 2 II Primary Decomposition p T Example 2.3. (i) For the nilradical of R we have N(R) := (0) = p2Spec(R) p. (ii) For a prime number p 2 we see pT (pn) = p(0) = (0) 6= (p) = T p(pn). Z n2N n2N This shows that we cannot omit the finiteness of I in Lemma 2.2 (iii). In some sence the concept of prime ideals corresponds to those of prime elements. Now we introduce ideals that – in a similar way – correspond to powers of prime elements. Definition 2.4. Let q ( R be an ideal such that p for x; y 2 R with xy 2 q and x2 = q we have y 2 q: p Then q is called primary ideal of R. In this case one says that q is p-primary if p = q. p Remark: In Proposition 2.8 we will see that for a primary ideal q the radical q ⊆ R is a prime ideal. Remark 2.5. An ideal q ⊆ R is primary if and only if every zero-divisor of R=q is nilpotent. Example 2.6. (i) Every prime ideal p is p-primary. n (ii) The primary ideals of Z are (0) and (p ) where p is a prime number and n ≥ 1. (iii) Consider the ideal q := (X; Y 2) ⊆ R := K[X; Y ] where K denotes a field. Since we have R=q =∼ K[Y ]=(Y 2) all zero-divisors of R=q are multiples of Y hence nilpotent. p 2 So q ⊆ R is primary. But we have p := q = (X; Y ) and p ( q ( p. This shows that a primary ideal need not be a power of a prime ideal. (iv) Conversely not every power of a prime ideal is primary: Let R := K[X; Y; Z]=(XY −Z2) and p := (X;¯ Z¯). Then p is prime since R=p =∼ K[Y ], but p 2.2 X¯ · Y¯ = Z¯2 2 p2 and X¯ 2 = p2; Y¯ 2 = p2 = p: So p2 is not primary although it is a power of a prime ideal. We have seen that in general the powers of prime ideals need not be primary.
Recommended publications
  • Dimension Theory and Systems of Parameters
    Dimension theory and systems of parameters Krull's principal ideal theorem Our next objective is to study dimension theory in Noetherian rings. There was initially amazement that the results that follow hold in an arbitrary Noetherian ring. Theorem (Krull's principal ideal theorem). Let R be a Noetherian ring, x 2 R, and P a minimal prime of xR. Then the height of P ≤ 1. Before giving the proof, we want to state a consequence that appears much more general. The following result is also frequently referred to as Krull's principal ideal theorem, even though no principal ideals are present. But the heart of the proof is the case n = 1, which is the principal ideal theorem. This result is sometimes called Krull's height theorem. It follows by induction from the principal ideal theorem, although the induction is not quite straightforward, and the converse also needs a result on prime avoidance. Theorem (Krull's principal ideal theorem, strong version, alias Krull's height theorem). Let R be a Noetherian ring and P a minimal prime ideal of an ideal generated by n elements. Then the height of P is at most n. Conversely, if P has height n then it is a minimal prime of an ideal generated by n elements. That is, the height of a prime P is the same as the least number of generators of an ideal I ⊆ P of which P is a minimal prime. In particular, the height of every prime ideal P is at most the number of generators of P , and is therefore finite.
    [Show full text]
  • CRITERIA for FLATNESS and INJECTIVITY 3 Ring of R
    CRITERIA FOR FLATNESS AND INJECTIVITY NEIL EPSTEIN AND YONGWEI YAO Abstract. Let R be a commutative Noetherian ring. We give criteria for flatness of R-modules in terms of associated primes and torsion-freeness of certain tensor products. This allows us to develop a criterion for regularity if R has characteristic p, or more generally if it has a locally contracting en- domorphism. Dualizing, we give criteria for injectivity of R-modules in terms of coassociated primes and (h-)divisibility of certain Hom-modules. Along the way, we develop tools to achieve such a dual result. These include a careful analysis of the notions of divisibility and h-divisibility (including a localization result), a theorem on coassociated primes across a Hom-module base change, and a local criterion for injectivity. 1. Introduction The most important classes of modules over a commutative Noetherian ring R, from a homological point of view, are the projective, flat, and injective modules. It is relatively easy to check whether a module is projective, via the well-known criterion that a module is projective if and only if it is locally free. However, flatness and injectivity are much harder to determine. R It is well-known that an R-module M is flat if and only if Tor1 (R/P,M)=0 for all prime ideals P . For special classes of modules, there are some criteria for flatness which are easier to check. For example, a finitely generated module is flat if and only if it is projective. More generally, there is the following Local Flatness Criterion, stated here in slightly simplified form (see [Mat86, Section 22] for a self-contained proof): Theorem 1.1 ([Gro61, 10.2.2]).
    [Show full text]
  • 23. Dimension Dimension Is Intuitively Obvious but Surprisingly Hard to Define Rigorously and to Work With
    58 RICHARD BORCHERDS 23. Dimension Dimension is intuitively obvious but surprisingly hard to define rigorously and to work with. There are several different concepts of dimension • It was at first assumed that the dimension was the number or parameters something depended on. This fell apart when Cantor showed that there is a bijective map from R ! R2. The Peano curve is a continuous surjective map from R ! R2. • The Lebesgue covering dimension: a space has Lebesgue covering dimension at most n if every open cover has a refinement such that each point is in at most n + 1 sets. This does not work well for the spectrums of rings. Example: dimension 2 (DIAGRAM) no point in more than 3 sets. Not trivial to prove that n-dim space has dimension n. No good for commutative algebra as A1 has infinite Lebesgue covering dimension, as any finite number of non-empty open sets intersect. • The "classical" definition. Definition 23.1. (Brouwer, Menger, Urysohn) A topological space has dimension ≤ n (n ≥ −1) if all points have arbitrarily small neighborhoods with boundary of dimension < n. The empty set is the only space of dimension −1. This definition is mostly used for separable metric spaces. Rather amazingly it also works for the spectra of Noetherian rings, which are about as far as one can get from separable metric spaces. • Definition 23.2. The Krull dimension of a topological space is the supre- mum of the numbers n for which there is a chain Z0 ⊂ Z1 ⊂ ::: ⊂ Zn of n + 1 irreducible subsets. DIAGRAM pt ⊂ curve ⊂ A2 For Noetherian topological spaces the Krull dimension is the same as the Menger definition, but for non-Noetherian spaces it behaves badly.
    [Show full text]
  • UC Berkeley UC Berkeley Previously Published Works
    UC Berkeley UC Berkeley Previously Published Works Title Operator bases, S-matrices, and their partition functions Permalink https://escholarship.org/uc/item/31n0p4j4 Journal Journal of High Energy Physics, 2017(10) ISSN 1126-6708 Authors Henning, B Lu, X Melia, T et al. Publication Date 2017-10-01 DOI 10.1007/JHEP10(2017)199 Peer reviewed eScholarship.org Powered by the California Digital Library University of California Published for SISSA by Springer Received: July 7, 2017 Accepted: October 6, 2017 Published: October 27, 2017 Operator bases, S-matrices, and their partition functions JHEP10(2017)199 Brian Henning,a Xiaochuan Lu,b Tom Meliac;d;e and Hitoshi Murayamac;d;e aDepartment of Physics, Yale University, New Haven, Connecticut 06511, U.S.A. bDepartment of Physics, University of California, Davis, California 95616, U.S.A. cDepartment of Physics, University of California, Berkeley, California 94720, U.S.A. dTheoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, California 94720, U.S.A. eKavli Institute for the Physics and Mathematics of the Universe (WPI), Todai Institutes for Advanced Study, University of Tokyo, Kashiwa 277-8583, Japan E-mail: [email protected], [email protected], [email protected], [email protected] Abstract: Relativistic quantum systems that admit scattering experiments are quan- titatively described by effective field theories, where S-matrix kinematics and symmetry considerations are encoded in the operator spectrum of the EFT. In this paper we use the S-matrix to derive the structure of the EFT operator basis, providing complementary de- scriptions in (i) position space utilizing the conformal algebra and cohomology and (ii) mo- mentum space via an algebraic formulation in terms of a ring of momenta with kinematics implemented as an ideal.
    [Show full text]
  • Notes on Irreducible Ideals
    BULL. AUSTRAL. MATH. SOC. I3AI5, I3H05, I 4H2O VOL. 31 (1985), 321-324. NOTES ON IRREDUCIBLE IDEALS DAVID J. SMITH Every ideal of a Noetherian ring may be represented as a finite intersection of primary ideals. Each primary ideal may be decomposed as an irredundant intersection of irreducible ideals. It is shown that in the case that Q is an Af-primary ideal of a local ring (if, M) satisfying the condition that Q : M = Q + M where s is the index of Q , then all irreducible components of Q have index s . (Q is "index- unmixed" .) This condition is shown to hold in the case that Q is a power of the maximal ideal of a regular local ring, and also in other cases as illustrated by examples. Introduction Let i? be a commutative ring with identity. An ideal J of R is irreducible if it is not a proper intersection of any two ideals of if . A discussion of some elementary properties of irreducible ideals is found in Zariski and Samuel [4] and Grobner [!]• If i? is Noetherian then every ideal of if has an irredundant representation as a finite intersection of irreducible ideals, and every irreducible ideal is primary. It is properties of representations of primary ideals as intersections of irreducible ideals which will be discussed here. We assume henceforth that if is Noetherian. Received 25 October 1981*. Copyright Clearance Centre, Inc. Serial-fee code: OOOl*-9727/85 $A2.00 + 0.00. 32 1 Downloaded from https://www.cambridge.org/core. IP address: 170.106.33.22, on 24 Sep 2021 at 06:01:50, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms.
    [Show full text]
  • Commutative Algebra
    Commutative Algebra Andrew Kobin Spring 2016 / 2019 Contents Contents Contents 1 Preliminaries 1 1.1 Radicals . .1 1.2 Nakayama's Lemma and Consequences . .4 1.3 Localization . .5 1.4 Transcendence Degree . 10 2 Integral Dependence 14 2.1 Integral Extensions of Rings . 14 2.2 Integrality and Field Extensions . 18 2.3 Integrality, Ideals and Localization . 21 2.4 Normalization . 28 2.5 Valuation Rings . 32 2.6 Dimension and Transcendence Degree . 33 3 Noetherian and Artinian Rings 37 3.1 Ascending and Descending Chains . 37 3.2 Composition Series . 40 3.3 Noetherian Rings . 42 3.4 Primary Decomposition . 46 3.5 Artinian Rings . 53 3.6 Associated Primes . 56 4 Discrete Valuations and Dedekind Domains 60 4.1 Discrete Valuation Rings . 60 4.2 Dedekind Domains . 64 4.3 Fractional and Invertible Ideals . 65 4.4 The Class Group . 70 4.5 Dedekind Domains in Extensions . 72 5 Completion and Filtration 76 5.1 Topological Abelian Groups and Completion . 76 5.2 Inverse Limits . 78 5.3 Topological Rings and Module Filtrations . 82 5.4 Graded Rings and Modules . 84 6 Dimension Theory 89 6.1 Hilbert Functions . 89 6.2 Local Noetherian Rings . 94 6.3 Complete Local Rings . 98 7 Singularities 106 7.1 Derived Functors . 106 7.2 Regular Sequences and the Koszul Complex . 109 7.3 Projective Dimension . 114 i Contents Contents 7.4 Depth and Cohen-Macauley Rings . 118 7.5 Gorenstein Rings . 127 8 Algebraic Geometry 133 8.1 Affine Algebraic Varieties . 133 8.2 Morphisms of Affine Varieties . 142 8.3 Sheaves of Functions .
    [Show full text]
  • Arxiv:1012.0864V3 [Math.AG] 14 Oct 2013 Nbt Emtyadpr Ler.Ltu Mhsz H Olwn Th Following the Emphasize Us H Let Results Cit.: His Algebra
    ORLOV SPECTRA: BOUNDS AND GAPS MATTHEW BALLARD, DAVID FAVERO, AND LUDMIL KATZARKOV Abstract. The Orlov spectrum is a new invariant of a triangulated category. It was intro- duced by D. Orlov building on work of A. Bondal-M. van den Bergh and R. Rouquier. The supremum of the Orlov spectrum of a triangulated category is called the ultimate dimension. In this work, we study Orlov spectra of triangulated categories arising in mirror symmetry. We introduce the notion of gaps and outline their geometric significance. We provide the first large class of examples where the ultimate dimension is finite: categories of singular- ities associated to isolated hypersurface singularities. Similarly, given any nonzero object in the bounded derived category of coherent sheaves on a smooth Calabi-Yau hypersurface, we produce a new generator by closing the object under a certain monodromy action and uniformly bound this new generator’s generation time. In addition, we provide new upper bounds on the generation times of exceptional collections and connect generation time to braid group actions to provide a lower bound on the ultimate dimension of the derived Fukaya category of a symplectic surface of genus greater than one. 1. Introduction The spectrum of a triangulated category was introduced by D. Orlov in [39], building on work of A. Bondal, R. Rouquier, and M. van den Bergh, [44] [11]. This categorical invariant, which we shall call the Orlov spectrum, is simply a list of non-negative numbers. Each number is the generation time of an object in the triangulated category. Roughly, the generation time of an object is the necessary number of exact triangles it takes to build the category using this object.
    [Show full text]
  • Commutative Algebra Ii, Spring 2019, A. Kustin, Class Notes
    COMMUTATIVE ALGEBRA II, SPRING 2019, A. KUSTIN, CLASS NOTES 1. REGULAR SEQUENCES This section loosely follows sections 16 and 17 of [6]. Definition 1.1. Let R be a ring and M be a non-zero R-module. (a) The element r of R is regular on M if rm = 0 =) m = 0, for m 2 M. (b) The elements r1; : : : ; rs (of R) form a regular sequence on M, if (i) (r1; : : : ; rs)M 6= M, (ii) r1 is regular on M, r2 is regular on M=(r1)M, ::: , and rs is regular on M=(r1; : : : ; rs−1)M. Example 1.2. The elements x1; : : : ; xn in the polynomial ring R = k[x1; : : : ; xn] form a regular sequence on R. Example 1.3. In general, order matters. Let R = k[x; y; z]. The elements x; y(1 − x); z(1 − x) of R form a regular sequence on R. But the elements y(1 − x); z(1 − x); x do not form a regular sequence on R. Lemma 1.4. If M is a finitely generated module over a Noetherian local ring R, then every regular sequence on M is a regular sequence in any order. Proof. It suffices to show that if x1; x2 is a regular sequence on M, then x2; x1 is a regular sequence on M. Assume x1; x2 is a regular sequence on M. We first show that x2 is regular on M. If x2m = 0, then the hypothesis that x1; x2 is a regular sequence on M guarantees that m 2 x1M; thus m = x1m1 for some m1.
    [Show full text]
  • NOTES in COMMUTATIVE ALGEBRA: PART 1 1. Results/Definitions Of
    NOTES IN COMMUTATIVE ALGEBRA: PART 1 KELLER VANDEBOGERT 1. Results/Definitions of Ring Theory It is in this section that a collection of standard results and definitions in commutative ring theory will be presented. For the rest of this paper, any ring R will be assumed commutative with identity. We shall also use "=" and "∼=" (isomorphism) interchangeably, where the context should make the meaning clear. 1.1. The Basics. Definition 1.1. A maximal ideal is any proper ideal that is not con- tained in any strictly larger proper ideal. The set of maximal ideals of a ring R is denoted m-Spec(R). Definition 1.2. A prime ideal p is such that for any a, b 2 R, ab 2 p implies that a or b 2 p. The set of prime ideals of R is denoted Spec(R). p Definition 1.3. The radical of an ideal I, denoted I, is the set of a 2 R such that an 2 I for some positive integer n. Definition 1.4. A primary ideal p is an ideal such that if ab 2 p and a2 = p, then bn 2 p for some positive integer n. In particular, any maximal ideal is prime, and the radical of a pri- mary ideal is prime. Date: September 3, 2017. 1 2 KELLER VANDEBOGERT Definition 1.5. The notation (R; m; k) shall denote the local ring R which has unique maximal ideal m and residue field k := R=m. Example 1.6. Consider the set of smooth functions on a manifold M.
    [Show full text]
  • Commutative Ideal Theory Without Finiteness Conditions: Completely Irreducible Ideals
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 358, Number 7, Pages 3113–3131 S 0002-9947(06)03815-3 Article electronically published on March 1, 2006 COMMUTATIVE IDEAL THEORY WITHOUT FINITENESS CONDITIONS: COMPLETELY IRREDUCIBLE IDEALS LASZLO FUCHS, WILLIAM HEINZER, AND BRUCE OLBERDING Abstract. An ideal of a ring is completely irreducible if it is not the intersec- tion of any set of proper overideals. We investigate the structure of completely irrreducible ideals in a commutative ring without finiteness conditions. It is known that every ideal of a ring is an intersection of completely irreducible ideals. We characterize in several ways those ideals that admit a representation as an irredundant intersection of completely irreducible ideals, and we study the question of uniqueness of such representations. We characterize those com- mutative rings in which every ideal is an irredundant intersection of completely irreducible ideals. Introduction Let R denote throughout a commutative ring with 1. An ideal of R is called irreducible if it is not the intersection of two proper overideals; it is called completely irreducible if it is not the intersection of any set of proper overideals. Our goal in this paper is to examine the structure of completely irreducible ideals of a commutative ring on which there are imposed no finiteness conditions. Other recent papers that address the structure and ideal theory of rings without finiteness conditions include [3], [4], [8], [10], [14], [15], [16], [19], [25], [26]. AproperidealA of R is completely irreducible if and only if there is an element x ∈ R such that A is maximal with respect to not containing x.
    [Show full text]
  • Commutative Algebra I
    Commutative Algebra I Craig Huneke 1 June 27, 2012 1A compilation of two sets of notes at the University of Kansas; one in the Spring of 2002 by ?? and the other in the Spring of 2007 by Branden Stone. These notes have been typed by Alessandro De Stefani and Branden Stone. Contents 1 Rings, Ideals, and Maps1 1 Notation and Examples.......................1 2 Homomorphisms and Isomorphisms.................2 3 Ideals and Quotient Rings......................3 4 Prime Ideals..............................6 5 Unique Factorization Domain.................... 13 2 Modules 19 1 Notation and Examples....................... 19 2 Submodules and Maps........................ 20 3 Tensor Products........................... 23 4 Operations on Modules....................... 29 3 Localization 33 1 Notation and Examples....................... 33 2 Ideals and Localization........................ 36 3 UFD's and Localization....................... 40 4 Chain Conditions 44 1 Noetherian Rings........................... 44 2 Noetherian Modules......................... 47 3 Artinian Rings............................ 49 5 Primary Decomposition 54 1 Definitions and Examples...................... 54 2 Primary Decomposition....................... 55 6 Integral Closure 62 1 Definitions and Notation....................... 62 2 Going-Up............................... 64 3 Normalization and Nullstellensatz.................. 67 4 Going-Down.............................. 71 5 Examples............................... 74 CONTENTS iii 7 Krull's Theorems and Dedekind Domains 77
    [Show full text]
  • Primary Decomposition
    + Primary Decomposition The decomposition of an ideal into primary ideals is a traditional pillar of ideal theory. It provides the algebraic foundation for decomposing an algebraic variety into its irreducible components-although it is only fair to point out that the algebraic picture is more complicated than naive geometry would suggest. From another point of view primary decomposition provides a gen- eralization of the factorization of an integer as a product of prime-powers. In the modern treatment, with its emphasis on localization, primary decomposition is no longer such a central tool in the theory. It is still, however, of interest in itself and in this chapter we establish the classical uniqueness theorems. The prototypes of commutative rings are z and the ring of polynomials kfxr,..., x,] where k is a field; both these areunique factorization domains. This is not true of arbitrary commutative rings, even if they are integral domains (the classical example is the ring Z[\/=1, in which the element 6 has two essentially distinct factorizations, 2.3 and it + r/-S;(l - /=)). However, there is a generalized form of "unique factorization" of ideals (not of elements) in a wide class of rings (the Noetherian rings). A prime ideal in a ring A is in some sense a generalization of a prime num- ber. The corresponding generalization of a power of a prime number is a primary ideal. An ideal q in a ring A is primary if q * A and. if xy€q => eitherxeqory" eqforsomen > 0. In other words, q is primary o Alq * 0 and every zero-divisor in l/q is nilpotent.
    [Show full text]