Integral Domains, Modules and Algebraic Integers Section 2 Hilary Term 2014
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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. -
Solutions of Some Exercises
Solutions of Some Exercises 1.3. n ∈ Let Specmax(R) and consider the homomorphism ϕ: R[x] → R/n,f→ f(0) + n. m ∩m n n ∈ The kernel of ϕ is a maximal ideal of R[x], and R = ,so Specrab(R). 2.5. (a)ThatS generates A means that for every element f ∈ A there exist finitely many elements f1,...,fm ∈ S and a polynomial F ∈ K[T1,...,Tm]inm indeterminates such that f = F (f1,...,fm). Let n P1,P2 ∈ K be points with f(P1) = f(P2). Then F (f1(P1),...,fm(P1)) = F (f1(P2),...,fn(P2)), so fi(P1) = fi(P2) for at least one i. This yields part (a). (b) Consider the polynomial ring B := K[x1,...,xn,y1,...,yn]in2n inde- terminates. Polynomials from B define functions Kn × Kn → K.For f ∈ K[x1,...,xn], define Δf := f(x1,...,xn) − f(y1,...,yn) ∈ B. n So for P1,P2 ∈ K we have Δf(P1,P2)=f(P1) − f(P2). Consider the ideal | ∈ ⊆ I := (Δf f A)B B. G. Kemper, A Course in Commutative Algebra, Graduate Texts 217 in Mathematics 256, DOI 10.1007/978-3-642-03545-6, c Springer-Verlag Berlin Heidelberg 2011 218 Solutions of exercises for Chapter 5 By Hilbert’s basis theorem (Corollary 2.13), B is Noetherian, so by Theorem 2.9 there exist f1,...,fm ∈ A such that I =(Δf1,...,Δfm)B . We claim that S := {f1,...,fm} is A-separating. For showing this, take n two points P1 and P2 in K and assume that there exists f ∈ A with f(P1) = f(P2). -
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. -
Integral Closures of Ideals and Rings Irena Swanson
Integral closures of ideals and rings Irena Swanson ICTP, Trieste School on Local Rings and Local Study of Algebraic Varieties 31 May–4 June 2010 I assume some background from Atiyah–MacDonald [2] (especially the parts on Noetherian rings, primary decomposition of ideals, ring spectra, Hilbert’s Basis Theorem, completions). In the first lecture I will present the basics of integral closure with very few proofs; the proofs can be found either in Atiyah–MacDonald [2] or in Huneke–Swanson [13]. Much of the rest of the material can be found in Huneke–Swanson [13], but the lectures contain also more recent material. Table of contents: Section 1: Integral closure of rings and ideals 1 Section 2: Integral closure of rings 8 Section 3: Valuation rings, Krull rings, and Rees valuations 13 Section 4: Rees algebras and integral closure 19 Section 5: Computation of integral closure 24 Bibliography 28 1 Integral closure of rings and ideals (How it arises, monomial ideals and algebras) Integral closure of a ring in an overring is a generalization of the notion of the algebraic closure of a field in an overfield: Definition 1.1 Let R be a ring and S an R-algebra containing R. An element x S is ∈ said to be integral over R if there exists an integer n and elements r1,...,rn in R such that n n 1 x + r1x − + + rn 1x + rn =0. ··· − This equation is called an equation of integral dependence of x over R (of degree n). The set of all elements of S that are integral over R is called the integral closure of R in S. -
9. Gauss Lemma Obviously It Would Be Nice to Have Some More General Methods of Proving That a Given Polynomial Is Irreducible. T
9. Gauss Lemma Obviously it would be nice to have some more general methods of proving that a given polynomial is irreducible. The first is rather beau- tiful and due to Gauss. The basic idea is as follows. Suppose we are given a polynomial with integer coefficients. Then it is natural to also consider this polynomial over the rationals. Note that it is much easier to prove that this polynomial is irreducible over the integers than it is to prove that it is irreducible over the rationals. For example it is clear that x2 − 2 is irreducible overp the integers. In fact it is irreducible over the rationals as well, that is, 2 is not a rational number. First some definitions. Definition 9.1. Let R be a commutative ring and let a1; a2; : : : ; ak be a sequence of elements of R. The gcd of a1; a2; : : : ; ak is an element d 2 R such that (1) djai, for all 1 ≤ i ≤ k. 0 0 (2) If d jai, for all 1 ≤ i ≤ k, then d jd. Lemma 9.2. Let R be a UFD. Then the gcd of any sequence a1; a2; : : : ; ak of elements of R exists. Proof. There are two obvious ways to proceed. The first is to take a common factorisation of each ai into a product of powers of primes, as in the case k = 2. The second is to recursively construct the gcd, by setting di to be the gcd of di−1 and ai and taking d1 = a1. In this case d = dk will be a gcd for the whole sequence a1; a2; : : : ; ak. -
Finiteness and Homological Conditions in Commutative Group Rings
XXXX, 1–15 © De Gruyter YYYY Finiteness and Homological Conditions in Commutative Group Rings Sarah Glaz and Ryan Schwarz Abstract. This article surveys the known results for several related families of ring properties in the context of commutative group rings. These properties include finiteness conditions, homological conditions, and conditions that connect these two families. We briefly survey the classical results, highlight the recent progress, and point out open problems and possible future directions of investigation in these areas. Keywords. Group rings, Noetherian rings, coherent rings, finite conductor rings, weak global dimension, von Neumann regular rings, semihereditary rings, Prüfer conditions, zero divisors, PP rings, PF rings. AMS classification. 13B99, 13D05, 13E99, 13F05. In memory of James Brewer with respect and affection 1 Introduction Let R be a commutative ring with identity and let G be an abelian group written multi- plicatively. The group ring RG is the free R module on the elements of G with multi- P plication induced by G. An element x in RG has a unique expression: x = g2G xgg, where xg 2 R and all but finitely many xg are zero. With addition, multiplication, and scalar multiplication by elements of R defined analogously to the standard polynomial operations, RG becomes a commutative R algebra. Properties of the group ring RG, particularly in conjunction with questions of de- scent and ascent of these properties between R and RG, have been of interest for at least 70 years. In his book Commutative Semigroup Rings [14], Gilmer traces the be- ginning of a systematic interest in the nature of RG, for general rings R and groups G, to Higman’s article [28] published in 1940. -
NOTES on UNIQUE FACTORIZATION DOMAINS Alfonso Gracia-Saz, MAT 347
Unique-factorization domains MAT 347 NOTES ON UNIQUE FACTORIZATION DOMAINS Alfonso Gracia-Saz, MAT 347 Note: These notes summarize the approach I will take to Chapter 8. You are welcome to read Chapter 8 in the book instead, which simply uses a different order, and goes in slightly different depth at different points. If you read the book, notice that I will skip any references to universal side divisors and Dedekind-Hasse norms. If you find any typos or mistakes, please let me know. These notes complement, but do not replace lectures. Last updated on January 21, 2016. Note 1. Through this paper, we will assume that all our rings are integral domains. R will always denote an integral domains, even if we don't say it each time. Motivation: We know that every integer number is the product of prime numbers in a unique way. Sort of. We just believed our kindergarden teacher when she told us, and we omitted the fact that it needed to be proven. We want to prove that this is true, that something similar is true in the ring of polynomials over a field. More generally, in which domains is this true? In which domains does this fail? 1 Unique-factorization domains In this section we want to define what it means that \every" element can be written as product of \primes" in a \unique" way (as we normally think of the integers), and we want to see some examples where this fails. It will take us a few definitions. Definition 2. Let a; b 2 R. -
Algebraic Number Theory Summary of Notes
Algebraic Number Theory summary of notes Robin Chapman 3 May 2000, revised 28 March 2004, corrected 4 January 2005 This is a summary of the 1999–2000 course on algebraic number the- ory. Proofs will generally be sketched rather than presented in detail. Also, examples will be very thin on the ground. I first learnt algebraic number theory from Stewart and Tall’s textbook Algebraic Number Theory (Chapman & Hall, 1979) (latest edition retitled Algebraic Number Theory and Fermat’s Last Theorem (A. K. Peters, 2002)) and these notes owe much to this book. I am indebted to Artur Costa Steiner for pointing out an error in an earlier version. 1 Algebraic numbers and integers We say that α ∈ C is an algebraic number if f(α) = 0 for some monic polynomial f ∈ Q[X]. We say that β ∈ C is an algebraic integer if g(α) = 0 for some monic polynomial g ∈ Z[X]. We let A and B denote the sets of algebraic numbers and algebraic integers respectively. Clearly B ⊆ A, Z ⊆ B and Q ⊆ A. Lemma 1.1 Let α ∈ A. Then there is β ∈ B and a nonzero m ∈ Z with α = β/m. Proof There is a monic polynomial f ∈ Q[X] with f(α) = 0. Let m be the product of the denominators of the coefficients of f. Then g = mf ∈ Z[X]. Pn j Write g = j=0 ajX . Then an = m. Now n n−1 X n−1+j j h(X) = m g(X/m) = m ajX j=0 1 is monic with integer coefficients (the only slightly problematical coefficient n −1 n−1 is that of X which equals m Am = 1). -
Gaussian Integers
Gaussian integers 1 Units in Z[i] An element x = a + bi 2 Z[i]; a; b 2 Z is a unit if there exists y = c + di 2 Z[i] such that xy = 1: This implies 1 = jxj2jyj2 = (a2 + b2)(c2 + d2) But a2; b2; c2; d2 are non-negative integers, so we must have 1 = a2 + b2 = c2 + d2: This can happen only if a2 = 1 and b2 = 0 or a2 = 0 and b2 = 1. In the first case we obtain a = ±1; b = 0; thus x = ±1: In the second case, we have a = 0; b = ±1; this yields x = ±i: Since all these four elements are indeed invertible we have proved that U(Z[i]) = {±1; ±ig: 2 Primes in Z[i] An element x 2 Z[i] is prime if it generates a prime ideal, or equivalently, if whenever we can write it as a product x = yz of elements y; z 2 Z[i]; one of them has to be a unit, i.e. y 2 U(Z[i]) or z 2 U(Z[i]): 2.1 Rational primes p in Z[i] If we want to identify which elements of Z[i] are prime, it is natural to start looking at primes p 2 Z and ask if they remain prime when we view them as elements of Z[i]: If p = xy with x = a + bi; y = c + di 2 Z[i] then p2 = jxj2jyj2 = (a2 + b2)(c2 + d2): Like before, a2 + b2 and c2 + d2 are non-negative integers. Since p is prime, the integers that divide p2 are 1; p; p2: Thus there are three possibilities for jxj2 and jyj2 : 1. -
Wheels — on Division by Zero Jesper Carlstr¨Om
ISSN: 1401-5617 Wheels | On Division by Zero Jesper Carlstr¨om Research Reports in Mathematics Number 11, 2001 Department of Mathematics Stockholm University Electronic versions of this document are available at http://www.matematik.su.se/reports/2001/11 Date of publication: September 10, 2001 2000 Mathematics Subject Classification: Primary 16Y99, Secondary 13B30, 13P99, 03F65, 08A70. Keywords: fractions, quotients, localization. Postal address: Department of Mathematics Stockholm University S-106 91 Stockholm Sweden Electronic addresses: http://www.matematik.su.se [email protected] Wheels | On Division by Zero Jesper Carlstr¨om Department of Mathematics Stockholm University http://www.matematik.su.se/~jesper/ Filosofie licentiatavhandling Abstract We show how to extend any commutative ring (or semiring) so that di- vision by any element, including 0, is in a sense possible. The resulting structure is what is called a wheel. Wheels are similar to rings, but 0x = 0 does not hold in general; the subset fx j 0x = 0g of any wheel is a com- mutative ring (or semiring) and any commutative ring (or semiring) with identity can be described as such a subset of a wheel. The main goal of this paper is to show that the given axioms for wheels are natural and to clarify how valid identities for wheels relate to valid identities for commutative rings and semirings. Contents 1 Introduction 3 1.1 Why invent the wheel? . 3 1.2 A sketch . 4 2 Involution-monoids 7 2.1 Definitions and examples . 8 2.2 The construction of involution-monoids from commutative monoids 9 2.3 Insertion of the parent monoid . -
NOTES on CHARACTERISTIC P COMMUTATIVE ALGEBRA MARCH 3RD, 2017
NOTES ON CHARACTERISTIC p COMMUTATIVE ALGEBRA MARCH 3RD, 2017 KARL SCHWEDE Proposition 0.1. Suppose that R ⊆ S is an inclusion of Noetherian domains such that S ∼= R ⊕ M as R-modules. Then if S is strongly F -regular, so is R. e Proof. Choose 0 6= c 2 R. Since S is strongly F -regular, there exists a φ : F∗ S −! S such e ∼ that φ(F∗ c) = 1. Let ρ : S −! R be such that ρ(1S) = 1R (this exists since S = R ⊕ M). e e φ ρ e Then the composition F∗ R ⊂ F∗ S −! S −! R sends F∗ c to 1 which proves that R is strongly F -regular. Remark 0.2. The above is an open problem in characteristic zero for KLT singularities. Corollary 0.3. A direct summand of a regular ring in characteristic p > 0 is Cohen- Macaulay. The above is obvious if we are taking a finite local inclusion of local rings of the same dimension. It is not so obvious otherwise (indeed, it has perhaps only recently been discovered how to show that direct summands of regular rings in mixed characteristic are Cohen-Macaulay). Proposition 0.4. If R is an F -finite ring such that Rm is strongly F -regular for each maximal m 2 Spec R, then R is strongly F -regular. Proof. Obviously strongly F -regular rings are F -split (take c = 1) and so R is F -split. By e post composing with Frobenius splittings, if we have a map φ : F∗ R −! R which sends e F∗ c 7! 1, then we can replace e by a larger e. -
Rings of Differential Operators and Étale Homomorphisms G´Isli Másson
Rings of differential operators and ´etale homomorphisms by G´ısli M´asson B.S., University of Iceland (1985) Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 1991 c Massachusetts Institute of Technology 1991 Signature of Author .................................................. Department of Mathematics May 2, 1991 Certified by .......................................................... Michael Artin Professor of Mathematics Thesis Supervisor Accepted by .......................................................... Sigurdur Helgason Chairman, Departmental Graduate Committee Department of Mathematics Rings of differential operators and ´etale homomorphisms by G´ısli M´asson Submitted to the Department of Mathematics on May 2, 1991, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract Rings of differential operators over rings of krull dimension 1 were studied by Musson, Smith and Stafford, and Muhasky. In particular it was proved that if k is a field of characteristic zero, and A is a finitely generated reduced k-algebra, then (i) D(A), the ring of differential operators on A, has a minimal essential twosided ideal J , (ii) A contains a minimal essential left D(A)-submodule, J,and (iii) the algebras C(A):=A/J and H(A):=D(A)/J are finite dimensional vector spaces over k. In this case the algebras C(A)andH(A) were studied by Brown and Smith. We use ´etale homomorphisms to obtain similar results in a somewhat more general setting. While an example of Bernstein, Gelfand and Gelfand shows that the statements above do not hold for all reduced finitely generated k-algebras of higher krull dimension, we present sufficient conditions on commutative, noetherian and reduced rings so that analogues of the ideal J and module J can be constructed, and for which statements similar to (i) and (ii) hold.