DOCSLIB.ORG

Lecture23 Print

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lecture23 Print Last time For the rest of today, let R be a ring with 1. For A R,let A denote the smallest ideal containing A,called Ä p q the ideal generated by A,andisequalto I. I an ideal A I £Ñ An ideal I is principle if there is some a R for which I a . P “p q For example, in Z x , r s 2x, 3x x , but 2,x is not principle. p q“p q p q Note: 2,x is principle in Q x , because it contains a unit! p q r s Fact: If u is a unit, then ur r . p q“p q Some facts from last time: 1. An ideal I of R is equal to R if and only if I contains a unit. 2. If R is commutative, then AR RA RAR A .Further, “ “ “p q R is a field if and only if its only ideals are 0 and R. 3. If R is a field then any nonzero ring homomorphism from R into another ring is an injection. An ideal M in the ring R is a maximal ideal if M R and the ‰ only ideals containing M are M and R. More facts from last time: 1. In a ring with identity every proper ideal is contained in a maximal ideal. 2. Let R be a commutative ring. The ideal M is maximal if and only if R M is a field. { Generalizing the integers: Prime ideals Ring theory in Number theory. For example, read about the reduction homomorphism (p. 245), and its role in finding integer solutions to equations like x2 y2 3z2 or xn yn zn ` “ ` “ Definition Let R be a commutative ring. An ideal P is a prime ideal if P R ‰ and whenever ab P ,eithera P or b P . P P P Example:WhataretheprimeidealsinZ? Proposition Let R be a commutative ring. The ideal P is a prime ideal if and only if R P is an integral domain. { Corollary Let R be a commutative ring. Every maximal ideal of R is a prime ideal. The many kinds of rings Assume all rings R have 1 for a moment. We already know. Commutative rings: multiplication is commutative. Division rings: R 0 , is a group. p ´t u ˆq Fields: R 0 , is an abelian group. p ´t u ˆq Integral domains (or domains, or IDs): commutative and no zero divisors (cancellation works). New: the many kinds of (integral) domains. Principal ideal domains (PIDs): every ideal is principal. Unique factorization domains (UFDs): elements factor uniquely into primes. Euclidean domains (EDs): there’s a division (i.e. Euclidean) algorithm (you need a norm to determine remainders). You can show that Rings Comm. rings IDs UFDs PIDs EDs Fields t u à t u à t u à t u à t u à t u à t u Let R be a commutative integral domain with 1. Definition Let a, b R with b 0. P ‰ 1. a is a multiple of b if there exists x R with a bx.Wesay P “ b divides a, denoted b a. | 2. A greatest common divisor of a and b is a nonzero element d dividing a and b such that d a and d b implies d d. 1 | 1 | 1 | Well-defined?? “a” versus “the” Proposition Let a, b, d, d R. 1 P 1. Abusing parentheses: If the ideal generated by a, b is the t u same as the ideal generated by d,thend gcd a, b ,i.e. “ p q a, b d a, b d. p q“p qùñpq“ 2. Uniqueness: If d and d1 generate the same ideal, then d ud for some unit u R . 1 “ P ˆ In particular, if d and d1 are both greatest common divisors of a and b,thend ud. 1 “ Finding and using greatest common divisors The previous proposition said if a, b d then d gcd a, b . p q“p q “ p q Careful! Not every ideal is principal! So the converse is not always true! Ex: x, y in Q x, y . p q r s (Corollary) However, if R is a principal ideal domain (an integral domain where every ideal is principal), we have 1. the ideal a, b is the same as the ideal gcd a, b ,forany p q p p qq gcd a, b ,andso p q 2. since a, b ar bs r, s R ,thismeansthat p q“t ` | P u ar bs gcd a, b for some r, s R. ` “ p q P But what are r and s? To calculate, we need a division algorithm! Euclidean domains is where that happens, but that’s ok: every PID is a ED. Some more facts about PID’s Recall that a prime ideal P is an ideal satisfying if ab P ,thena P or b P . P P P Also, recall that maximal ideals are all prime ideals. Proposition In a PID, every non-zero prime ideal is a maximal ideal. Corollary If R is commutative and R x is a PID, then R is a field. r s.
Load more

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]
  • Gsm073-Endmatter.Pdf
    http://dx.doi.org/10.1090/gsm/073 Graduat e Algebra : Commutativ e Vie w This page intentionally left blank Graduat e Algebra : Commutativ e View Louis Halle Rowen Graduate Studies in Mathematics Volum e 73 KHSS^ K l|y|^| America n Mathematica l Societ y iSyiiU ^ Providence , Rhod e Islan d Contents Introduction xi List of symbols xv Chapter 0. Introduction and Prerequisites 1 Groups 2 Rings 6 Polynomials 9 Structure theories 12 Vector spaces and linear algebra 13 Bilinear forms and inner products 15 Appendix 0A: Quadratic Forms 18 Appendix OB: Ordered Monoids 23 Exercises - Chapter 0 25 Appendix 0A 28 Appendix OB 31 Part I. Modules Chapter 1. Introduction to Modules and their Structure Theory 35 Maps of modules 38 The lattice of submodules of a module 42 Appendix 1A: Categories 44 VI Contents Chapter 2. Finitely Generated Modules 51 Cyclic modules 51 Generating sets 52 Direct sums of two modules 53 The direct sum of any set of modules 54 Bases and free modules 56 Matrices over commutative rings 58 Torsion 61 The structure of finitely generated modules over a PID 62 The theory of a single linear transformation 71 Application to Abelian groups 77 Appendix 2A: Arithmetic Lattices 77 Chapter 3. Simple Modules and Composition Series 81 Simple modules 81 Composition series 82 A group-theoretic version of composition series 87 Exercises — Part I 89 Chapter 1 89 Appendix 1A 90 Chapter 2 94 Chapter 3 96 Part II. AfRne Algebras and Noetherian Rings Introduction to Part II 99 Chapter 4. Galois Theory of Fields 101 Field extensions 102 Adjoining
    [Show full text]
  • Ring (Mathematics) 1 Ring (Mathematics)
    Ring (mathematics) 1 Ring (mathematics) In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition (called the additive group of the ring) and a monoid under multiplication such that multiplication distributes over addition.a[›] In other words the ring axioms require that addition is commutative, addition and multiplication are associative, multiplication distributes over addition, each element in the set has an additive inverse, and there exists an additive identity. One of the most common examples of a ring is the set of integers endowed with its natural operations of addition and multiplication. Certain variations of the definition of a ring are sometimes employed, and these are outlined later in the article. Polynomials, represented here by curves, form a ring under addition The branch of mathematics that studies rings is known and multiplication. as ring theory. Ring theorists study properties common to both familiar mathematical structures such as integers and polynomials, and to the many less well-known mathematical structures that also satisfy the axioms of ring theory. The ubiquity of rings makes them a central organizing principle of contemporary mathematics.[1] Ring theory may be used to understand fundamental physical laws, such as those underlying special relativity and symmetry phenomena in molecular chemistry. The concept of a ring first arose from attempts to prove Fermat's last theorem, starting with Richard Dedekind in the 1880s. After contributions from other fields, mainly number theory, the ring notion was generalized and firmly established during the 1920s by Emmy Noether and Wolfgang Krull.[2] Modern ring theory—a very active mathematical discipline—studies rings in their own right.
    [Show full text]
  • Are Maximal Ideals? Solution. No Principal
    Math 403A Assignment 6. Due March 1, 2013. 1.(8.1) Which principal ideals in Z[x] are maximal ideals? Solution. No principal ideal (f(x)) is maximal. If f(x) is an integer n = 1, then (n, x) is a bigger ideal that is not the whole ring. If f(x) has positive degree, then ± take any prime number p that does not divide the leading coefficient of f(x). (p, f(x)) is a bigger ideal that is not the whole ring, since Z[x]/(p, f(x)) = Z/pZ[x]/(f(x)) is not the zero ring. 2.(8.2) Determine the maximal ideals in the following rings. (a). R R. × Solution. Since (a, b) is a unit in this ring if a and b are nonzero, a maximal ideal will contain no pair (a, 0), (0, b) with a and b nonzero. Hence, the maximal ideals are R 0 and 0 R. ×{ } { } × (b). R[x]/(x2). Solution. Every ideal in R[x] is principal, i.e., generated by one element f(x). The ideals in R[x] that contain (x2) are (x2) and (x), since if x2 R[x]f(x), then x2 = f(x)g(x), which 2 ∈ 2 2 means that f(x) divides x . Up to scalar multiples, the only divisors of x are x and x . (c). R[x]/(x2 3x + 2). − Solution. The nonunit divisors of x2 3x +2=(x 1)(x 2) are, up to scalar multiples, x2 3x + 2 and x 1 and x 2. The− maximal ideals− are− (x 1) and (x 2), since the quotient− of R[x] by− either of those− ideals is the field R.
    [Show full text]
  • Class Numbers of Number Rings: Existence and Computations
    CLASS NUMBERS OF NUMBER RINGS: EXISTENCE AND COMPUTATIONS GABRIEL GASTER Abstract. This paper gives a brief introduction to the theory of algebraic numbers, with an eye toward computing the class numbers of certain number rings. The author adapts the treatments of Madhav Nori (from [1]) and Daniel Marcus (from [2]). None of the following work is original, but the author does include some independent solutions of exercises given by Nori or Marcus. We assume familiarity with the basic theory of fields. 1. A Note on Notation Throughout, unless explicitly stated, R is assumed to be a commutative integral domain. As is customary, we write Frac(R) to denote the field of fractions of a domain. By Z, Q, R, C we denote the integers, rationals, reals, and complex numbers, respectively. For R a ring, R[x], read ‘R adjoin x’, is the polynomial ring in x with coefficients in R. 2. Integral Extensions In his lectures, Madhav Nori said algebraic number theory is misleadingly named. It is not the application of modern algebraic tools to number theory; it is the study of algebraic numbers. We will mostly, therefore, concern ourselves solely with number fields, that is, finite (hence algebraic) extensions of Q. We recall from our experience with linear algebra that much can be gained by slightly loosening the requirements on the structures in question: by studying mod- ules over a ring one comes across entire phenomena that are otherwise completely obscured when one studies vector spaces over a field. Similarly, here we adapt our understanding of algebraic extensions of a field to suitable extensions of rings, called ‘integral.’ Whereas Z is a ‘very nice’ ring—in some sense it is the nicest possible ring that is not a field—the integral extensions of Z, called ‘number rings,’ have surprisingly less structure.
    [Show full text]
  • Ideals and Class Groups of Number Fields
    Ideals and class groups of number fields A thesis submitted To Kent State University in partial Fulfillment of the requirements for the Degree of Master of Science by Minjiao Yang August, 2018 ○C Copyright All rights reserved Except for previously published materials Thesis written by Minjiao Yang B.S., Kent State University, 2015 M.S., Kent State University, 2018 Approved by Gang Yu , Advisor Andrew Tonge , Chair, Department of Mathematics Science James L. Blank , Dean, College of Arts and Science TABLE OF CONTENTS…………………………………………………………...….…...iii ACKNOWLEDGEMENTS…………………………………………………………....…...iv CHAPTER I. Introduction…………………………………………………………………......1 II. Algebraic Numbers and Integers……………………………………………......3 III. Rings of Integers…………………………………………………………….......9 Some basic properties…………………………………………………………...9 Factorization of algebraic integers and the unit group………………….……....13 Quadratic integers……………………………………………………………….17 IV. Ideals……..………………………………………………………………….......21 A review of ideals of commutative……………………………………………...21 Ideal theory of integer ring 풪푘…………………………………………………..22 V. Ideal class group and class number………………………………………….......28 Finiteness of 퐶푙푘………………………………………………………………....29 The Minkowski bound…………………………………………………………...32 Further remarks…………………………………………………………………..34 BIBLIOGRAPHY…………………………………………………………….………….......36 iii ACKNOWLEDGEMENTS I want to thank my advisor Dr. Gang Yu who has been very supportive, patient and encouraging throughout this tremendous and enchanting experience. Also, I want to thank my thesis committee members Dr. Ulrike Vorhauer and Dr. Stephen Gagola who help me correct mistakes I made in my thesis and provided many helpful advices. iv Chapter 1. Introduction Algebraic number theory is a branch of number theory which leads the way in the world of mathematics. It uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Concepts and results in algebraic number theory are very important in learning mathematics.
    [Show full text]
  • A Friendly Introduction to Ideal Class Groups and an Interesting Problem
    A Friendly Introduction to Ideal Class Groups and An Interesting Problem Sutirtha Datta,F.Y.B.Sc(CMI) July 2019 1 Introduction In this article, we will mainly deal with one of the many interesting topics of Algebraic Number Theory : Class Numbers ( Red Alert:Please don't drag CMI Alg-2 topic on conjugacy classes in a group here) and we will see a classic prob- lem known as "Gauss's class number problem for imaginary quadratic fields”, which will eventually lead us to a sweet theorem named Heegner Theorem (credit goes to Harold Stark as well) I assume the reader to know basic notions of ring,ideals,field,field extensions etc ,else they can just have a glance on my Galois Theory article.Still I will give definitions of basic notions of UFD, PID and so on below. 2 Some Preliminary Concepts 2.1 Notion of Integral Closure Before I give the definitions,I would like to ask the reader to realize the similarity between the notion of an element being algebraic "over" an algebraic extension and that of integral "over" an integral extension while going through this section. Consider R is a subring of the commutative ring S with 1 = 1S 2 R. Then S is integral over R if every s 2 S is integral over R Now what does being integral over some ring mean? Definition 2.1 (Integral Element). : An element s 2 S is integral over R if s is root of a monic polynomial in R[x]. Definition 2.2 (Integral Closure). : The integral closure of R in S is the set of elements of S that are integral over R.
    [Show full text]
  • The Different Ideal
    THE DIFFERENT IDEAL KEITH CONRAD 1. Introduction The discriminant of a number field K tells us which primes p in Z ramify in OK : the prime factors of the discriminant. However, the way we have seen how to compute the discriminant doesn't address the following themes: (a) determine which prime ideals in OK ramify (that is, which p in OK have e(pjp) > 1 rather than which p have e(pjp) > 1 for some p), (b) determine the multiplicity of a prime in the discriminant. (We only know the mul- tiplicity is positive for the ramified primes.) Example 1.1. Let K = Q(α), where α3 − α − 1 = 0. The polynomial T 3 − T − 1 has discriminant −23, which is squarefree, so OK = Z[α] and we can detect how a prime p 3 factors in OK by seeing how T − T − 1 factors in Fp[T ]. Since disc(OK ) = −23, only the prime 23 ramifies. Since T 3−T −1 ≡ (T −3)(T −10)2 mod 23, (23) = pq2. One prime over 23 has multiplicity 1 and the other has multiplicity 2. The discriminant tells us some prime over 23 ramifies, but not which ones ramify. Only q does. The discriminant of K is, by definition, the determinant of the matrix (TrK=Q(eiej)), where e1; : : : ; en is an arbitrary Z-basis of OK . By a finer analysis of the trace, we will construct an ideal in OK which is divisible precisely by the ramified primes in OK . This ideal is called the different ideal. (It is related to differentiation, hence the name I think.) In the case of Example 1.1, for instance, we will see that the different ideal is q, so the different singles out the particular prime over 23 that ramifies.
    [Show full text]
  • 1.1 Rings and Ideals
    1.1 Rings and Ideals A ring A is a set with + , such that • (1) (A, +) is an abelian group; (2) (A, ) is a semigroup; • (3) distributes over + on both sides. • 22 In this course all rings A are commutative, that is, (4) ( x,y A) x y = y x ∀ ∈ • • and have an identity element 1 (easily seen to be unique) (5) ( 1 A)( x A) 1 x = x 1 = x . ∃ ∈ ∀ ∈ • • 23 If 1=0 then A = 0 (easy to see), { } called the zero ring. Multiplication will be denoted by juxtaposition, and simple facts used without comment, such as ( x,y A) ∀ ∈ x 0 = 0 , ( x)y = x( y) = (xy) , − − − ( x)( y) = xy . − − 24 Call a subset S of a ring A a subring if (i) 1 S ; ∈ (ii) ( x,y S) x+y, xy, x S . ∀ ∈ − ∈ Condition (ii) is easily seen to be equivalent to (ii)′ ( x,y S) x y, xy S . ∀ ∈ − ∈ 25 Note: In other contexts authors replace the condition 1 S by S = (which is ∈ 6 ∅ not equivalent!). Examples: (1) Z is the only subring of Z . (2) Z is a subring of Q , which is a subring of R , which is a subring of C . 26 (3) Z[i] = a + bi a, b Z (i = √ 1) , { | ∈ } − the ring of Gaussian integers is a subring of C . (4) Z = 0, 1,...,n 1 n { − } with addition and multiplication mod n . (Alternatively Zn may be defined to be the quotient ring Z/nZ , defined below). 27 (5) R any ring, x an indeterminate. Put R[[x]] = a +a x+a x2 +..
    [Show full text]
  • CHAPTER 2 RING FUNDAMENTALS 2.1 Basic Definitions and Properties
    page 1 of Chapter 2 CHAPTER 2 RING FUNDAMENTALS 2.1 Basic Definitions and Properties 2.1.1 Definitions and Comments A ring R is an abelian group with a multiplication operation (a, b) → ab that is associative and satisfies the distributive laws: a(b+c)=ab+ac and (a + b)c = ab + ac for all a, b, c ∈ R. We will always assume that R has at least two elements,including a multiplicative identity 1 R satisfying a1R =1Ra = a for all a in R. The multiplicative identity is often written simply as 1,and the additive identity as 0. If a, b,and c are arbitrary elements of R,the following properties are derived quickly from the definition of a ring; we sketch the technique in each case. (1) a0=0a =0 [a0+a0=a(0+0)=a0; 0a +0a =(0+0)a =0a] (2) (−a)b = a(−b)=−(ab)[0=0b =(a+(−a))b = ab+(−a)b,so ( −a)b = −(ab); similarly, 0=a0=a(b +(−b)) = ab + a(−b), so a(−b)=−(ab)] (3) (−1)(−1) = 1 [take a =1,b= −1 in (2)] (4) (−a)(−b)=ab [replace b by −b in (2)] (5) a(b − c)=ab − ac [a(b +(−c)) = ab + a(−c)=ab +(−(ac)) = ab − ac] (6) (a − b)c = ac − bc [(a +(−b))c = ac +(−b)c)=ac − (bc)=ac − bc] (7) 1 = 0 [If 1 = 0 then for all a we have a = a1=a0 = 0,so R = {0},contradicting the assumption that R has at least two elements] (8) The multiplicative identity is unique [If 1 is another multiplicative identity then 1=11 =1] 2.1.2 Definitions and Comments If a and b are nonzero but ab = 0,we say that a and b are zero divisors;ifa ∈ R and for some b ∈ R we have ab = ba = 1,we say that a is a unit or that a is invertible.
    [Show full text]
  • Principal Ideal Domains
    MAT 511 - Fall 2015 Principle Ideal Domains Principal Ideal Domains This section of notes roughly follows Sections 8.1-8.2 in Dummit and Foote. Throughout this whole section, we assume that R is a commutative ring. Definition 55. Let R b a commutative ring and let a;b R with b , 0. 2 (1) a is said to be multiple of b if there exists an element x R with a = bx. In this case, b is 2 said to divide a or be a divisor of a, written b a. j (2)A greatest common divisor of a and b is a nonzero element d such that (a) d a and d b, and j j (b) if d a and d b, then d d. 0 j 0 j 0 j A greatest common divisor of a and b will be denoted gcd(a;b) (or possibly (a;b)). Note 56. Note that b a in a ring R iff a (b) iff (a) (b). In particular, if d is any divisor of j 2 ⊆ both a and b, then (d) must contain both a and b, and hence must contain (a;b). Moreover, if d = gcd(a;b) iff (a;b) (d) and if (d ) is any principal ideal containing (a;b), then (d) (d ). ⊆ 0 ⊆ 0 The note above immediately proves the following result. Theorem 57. If a and b are nonzero elements in the commutative ring R such that (a;b) = (d), then d = gcd(a;b). Note 58. It is important to point out that the theorem above is giving us a sufficient condition, but it is not necessary.
    [Show full text]
  • Algebraic Number Theory
    Algebraic Number Theory George D. Torres Math 390C - Fall 2017 1 Number Fields 2 1.1 Norm, Trace, and Discriminant . .3 1.2 Algebraic Integers . .5 2 Dedekind Rings 7 2.1 Fractional Ideals and Unique Factorization . .8 2.2 The Ideal Class Group . 10 3 Geometry of Numbers 14 3.1 Lattices . 14 3.2 Minkowski’s Theorem . 17 3.3 Dirichlet’s Theorem . 20 4 Ramification and Decomposition of Primes 23 4.1 Prime ideals in Galois Extensions . 26 4.1.1 Decomposition and Inertia Groups . 27 4.1.2 The Abelian, Unramified Case . 28 5 Local Fields 29 5.1 Defining Local Fields . 30 5.2 Finite extensions of Qp ........................................... 32 5.2.1 Ramification in p-adic fields . 34 5.2.2 Krasner’s Lemma . 35 5.3 The Approximation Theorem and Ostrowski’s Theorem . 36 6 Analytic Methods in Number Fields 39 6.1 The Zeta Function of a Number Field . 40 6.2 Dirichlet Characters and L-Functions . 44 6.2.1 Quadratic Characters and Quadratic Fields . 46 7 Introduction to Global Class Field Theory 50 7.1 Moduli and the Ray Class Group . 50 7.2 The Artin Symbol . 51 ————————————————– These are lecture notes from Mirela Çiperiani’s Algebraic Number Theory course M390C given Fall 2017. The reader should be comfortable with the essential notions of commutative algebra, general field theory, and Galois theory. Please forward any typos to [email protected]. Last updated: February 13, 2019 1 1. Number Fields v A number field is a finite extension of Q. To be more precise, we’ll begin with some review of field extensions and algebraic elements.
    [Show full text]