On the Existence of Maximal Orders
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The Class Number One Problem for Imaginary Quadratic Fields
MODULAR CURVES AND THE CLASS NUMBER ONE PROBLEM JEREMY BOOHER Gauss found 9 imaginary quadratic fields with class number one, and in the early 19th century conjectured he had found all of them. It turns out he was correct, but it took until the mid 20th century to prove this. Theorem 1. Let K be an imaginary quadratic field whose ring of integers has class number one. Then K is one of p p p p p p p p Q(i); Q( −2); Q( −3); Q( −7); Q( −11); Q( −19); Q( −43); Q( −67); Q( −163): There are several approaches. Heegner [9] gave a proof in 1952 using the theory of modular functions and complex multiplication. It was dismissed since there were gaps in Heegner's paper and the work of Weber [18] on which it was based. In 1967 Stark gave a correct proof [16], and then noticed that Heegner's proof was essentially correct and in fact equiv- alent to his own. Also in 1967, Baker gave a proof using lower bounds for linear forms in logarithms [1]. Later, Serre [14] gave a new approach based on modular curve, reducing the class number + one problem to finding special points on the modular curve Xns(n). For certain values of n, it is feasible to find all of these points. He remarks that when \N = 24 An elliptic curve is obtained. This is the level considered in effect by Heegner." Serre says nothing more, and later writers only repeat this comment. This essay will present Heegner's argument, as modernized in Cox [7], then explain Serre's strategy. -
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. -
Multiplication Domains, Nagata Rings, and Kronecker Function Rings
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Algebra 319 (2008) 309–319 www.elsevier.com/locate/jalgebra Prüfer ∗-multiplication domains, Nagata rings, and Kronecker function rings Gyu Whan Chang Department of Mathematics, University of Incheon, Incheon 402-749, Republic of Korea Received 31 January 2007 Available online 30 October 2007 Communicated by Steven Dale Cutkosky Abstract Let D be an integrally closed domain, ∗ a star-operation on D, X an indeterminate over D,andN∗ = ∗ {f ∈ D[X]|(Af ) = D}.Forane.a.b. star-operation ∗1 on D,letKr(D, ∗1) be the Kronecker function ring of D with respect to ∗1. In this paper, we use ∗ todefineanewe.a.b. star-operation ∗c on D.Then ∗ [ ] = ∗ we prove that D is a Prüfer -multiplication domain if and only if D X N∗ Kr(D, c), if and only if Kr(D, ∗c) is a quotient ring of D[X], if and only if Kr(D, ∗c) is a flat D[X]-module, if and only if each ∗-linked overring of D is a Prüfer v-multiplication domain. This is a generalization of the following well- known fact that if D is a v-domain, then D is a Prüfer v-multiplication domain if and only if Kr(D, v) = [ ] [ ] [ ] D X Nv , if and only if Kr(D, v) is a quotient ring of D X , if and only if Kr(D, v) is a flat D X -module. © 2007 Elsevier Inc. All rights reserved. Keywords: (e.a.b.) ∗-operation; Prüfer ∗-multiplication domain; Nagata ring; Kronecker function ring 1. -
Divisibility Theory of Semi-Hereditary Rings
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 138, Number 12, December 2010, Pages 4231–4242 S 0002-9939(2010)10465-3 Article electronically published on July 9, 2010 DIVISIBILITY THEORY OF SEMI-HEREDITARY RINGS P. N. ANH´ AND M. SIDDOWAY (Communicated by Birge Huisgen-Zimmermann) Abstract. The semigroup of finitely generated ideals partially ordered by in- verse inclusion, i.e., the divisibility theory of semi-hereditary rings, is precisely described by semi-hereditary Bezout semigroups. A Bezout semigroup is a commutative monoid S with 0 such that the divisibility relation a|b ⇐⇒ b ∈ aS is a partial order inducing a distributive lattice on S with multiplication dis- tributive on both meets and joins, and for any a, b, d = a ∧ b ∈ S, a = da1 there is b1 ∈ S with a1 ∧ b1 =1,b= db1. S is semi-hereditary if for each a ∈ S there is e2 = e ∈ S with eS = a⊥ = {x ∈ S | ax =0}. The dictionary is therefore complete: abelian lattice-ordered groups and semi-hereditary Be- zout semigroups describe divisibility of Pr¨ufer (i.e., semi-hereditary) domains and semi-hereditary rings, respectively. The construction of a semi-hereditary Bezout ring with a pre-described semi-hereditary Bezout semigroup is inspired by Stone’s representation of Boolean algebras as rings of continuous functions and by Gelfand’s and Naimark’s analogous representation of commutative C∗- algebras. Introduction In considering the structure of rings, classical number theory suggests a careful study of the semigroup of divisibility, defined as the semigroup of principal (or more generally finitely generated) ideals partially ordered by reverse inclusion. -
Artinian Subrings of a Commutative Ring
transactions of the american mathematical society Volume 336, Number 1, March 1993 ARTINIANSUBRINGS OF A COMMUTATIVERING ROBERT GILMER AND WILLIAM HEINZER Abstract. Given a commutative ring R, we investigate the structure of the set of Artinian subrings of R . We also consider the family of zero-dimensional subrings of R. Necessary and sufficient conditions are given in order that every zero-dimensional subring of a ring be Artinian. We also consider closure properties of the set of Artinian subrings of a ring with respect to intersection or finite intersection, and the condition that the set of Artinian subrings of a ring forms a directed family. 1. Introduction Suppose R is a commutative ring with unity element. We consider here various properties of the family sf of Artinian subrings of R and the family Z of zero-dimensional subrings of R . We remark that the inclusion s? ç Z may be proper, even if R is Noetherian; for example, Corollary 3.4 implies that if K is an infinite field of positive characteristic, then the local principal ideal ring K[X]/(X2) contains a zero-dimensional subring that is not Artinian. Of course, if every subring of R were Noetherian, the families sf and Z would be identical. Thus one source of motivation for this work comes from papers such as [Gi, Wi, W2, GHi, GH3] that deal with Noetherian and zero- dimensional pairs of rings, hereditarily Noetherian rings, and hereditarily zero- dimensional rings. Another source of motivation is related to our work in [GH3], where we considered several problems concerning a direct product of zero-dimensional rings. -
Maximal Orders of Quaternions
Maximal Orders of Quaternions Alyson Deines Quaternion Algebras Maximal Orders of Quaternions Maximal Orders Norms, Traces, and Alyson Deines Integrality Local Fields and Hilbert Symbols December 8th Math 581b Final Project Uses A non-commutative analogue of Rings of Integers of Number Field Abstract Maximal Orders of Quaternions Alyson Deines Abstract Quaternion I will define maximal orders of quaternion algebras, give Algebras Maximal examples, discuss how they are similar to and differ from rings Orders of integers, and then I will discuss one of their applications . Norms, Traces, and Integrality Local Fields and Hilbert Symbols Uses Quaternion Algebras over Number Fields Maximal Orders of Definition Quaternions Alyson Deines Let F be a field and a; b 2 F . A quaternion algebra over F is an F -algebra B = a;b Quaternion F Algebras with basis Maximal 2 2 Orders i = a; j = b; and ij = −ji: Norms, Traces, and Integrality Examples: Local Fields −1;−1 and Hilbert = Symbols H R p Uses −1;−1 F = Q( 5), B = F 1;b ∼ Let F be any number field, F = M2(F ) via 1 0 0 b i 7! and j 7! 0 −1 1 0 Maximal Orders Maximal Orders of Quaternions Lattices Alyson Deines Let F be a number field with ring of integers R and let V Quaternion be a finite dimensional F -vector space. Algebras An R-lattice of V is a finitely generated R-submodule Maximal Orders O ⊂ V such that OF = V . Norms, Traces, and Integrality Orders Local Fields and Hilbert Let F be a number field with ring of integers R and let B Symbols be a finite dimensional F -algebra. -
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. -
A Brief History of Ring Theory
A Brief History of Ring Theory by Kristen Pollock Abstract Algebra II, Math 442 Loyola College, Spring 2005 A Brief History of Ring Theory Kristen Pollock 2 1. Introduction In order to fully define and examine an abstract ring, this essay will follow a procedure that is unlike a typical algebra textbook. That is, rather than initially offering just definitions, relevant examples will first be supplied so that the origins of a ring and its components can be better understood. Of course, this is the path that history has taken so what better way to proceed? First, it is important to understand that the abstract ring concept emerged from not one, but two theories: commutative ring theory and noncommutative ring the- ory. These two theories originated in different problems, were developed by different people and flourished in different directions. Still, these theories have much in com- mon and together form the foundation of today's ring theory. Specifically, modern commutative ring theory has its roots in problems of algebraic number theory and algebraic geometry. On the other hand, noncommutative ring theory originated from an attempt to expand the complex numbers to a variety of hypercomplex number systems. 2. Noncommutative Rings We will begin with noncommutative ring theory and its main originating ex- ample: the quaternions. According to Israel Kleiner's article \The Genesis of the Abstract Ring Concept," [2]. these numbers, created by Hamilton in 1843, are of the form a + bi + cj + dk (a; b; c; d 2 R) where addition is through its components 2 2 2 and multiplication is subject to the relations i =pj = k = ijk = −1. -
Recent Developments on the Kronecker Function Rings Theory
UNIVERSITA` DEGLI STUDI ROMA TRE FACOLTA` DI SCIENZE M.F.N. Tesi di Laurea Specialistica in Matematica SINTESI presentata da Elisa Di Gloria Recent developments on the Kronecker function rings Theory Relatore Prof. Marco Fontana Il Candidato Il Relatore ANNO ACCADEMICO 2009 - 2010 II Sessione - Ottobre 2010 Classificazione AMS : 13F05; 13G05; 13A15; 13A18 Parole Chiave : Anelli di Valutazione, Domini di Pr¨ufer,Star operazioni, Anello di Nagata, Anello delle funzioni di Kronecker Chapter 1 1.1 b-operation and integral closure Our work starts from the definition of the b-operation, a (semi)star operation on an integral domain D. Here, we start by giving some notation. Let D be an integral domain with quotient field K. Let F(D) be the set of all nonzero D-submodules of K and F(D) the nonzero fractionary ideals of D and, finally, let f (D) be the finitely generated D-submodules of K. Hence: f (D) ⊆ F(D) ⊆ F(D): Definition 1.1.1. If M is a D-module contained in K, the completion of M is the D-module \ Mf := MVλ: Vλ2S The module M is said to be complete if M = Mf. Remark 1.1.2. If D¯ denotes the integral closure of D in K and if we set M¯ := MD¯, then Mf¯ = Mf, where Mf¯ is the completion of the D¯-module M¯ . Proof: By definition, S is the set of all valuation overrings of D¯, hence: ¯ \ ¯ \ Mf = MVλ = MVλ = M:f Vλ2S Vλ2S It follows that the class of complete D¯-modules coincides with the class of com- plete D-modules. -
Hereditary Rings Integral Over Their Centers
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 102, 119-128 (1986) Hereditary Rings Integral over Their Centers ELLEN KIRKMAN AND JAMES KUZMANOVICH Department of Mathematics and Computer Science, Wake Forest University, Box 7311, Reynolda Station, Winston-Salem, North Carolina 27109 Communicated by P. M. Cohn Received October 1, 1984 Bergman [2] has completely characterized the center of a right hereditary ring; the center of a right hereditary ring is a Krull p. p. ring, and any Krull p. p. ring is the center of a right hereditary. While the center of a hereditary ring thus need not be hereditary, there are conditions which imply that the center is hereditary. Robson and Small [13] have shown that the center of a prime PI right hereditary ring is a Dedekind domain, and the center of a PI hereditary Noetherian ring is a finite direct sum of Dedekind domains. The center of a right hereditary PI ring need not be hereditary; Small and Wadsworth [16] have given an example of a PI right hereditary, right Noetherian ring whose center is not Noetherian or semihereditary. Jondrup [ 111 showed that a right hereditary ring which is module-finite over its center has a hereditary center. Chatters and Jondrup [S] showed that a PI right hereditary ring which is ring-finite over its cen- ter has a hereditary center. We shall prove that any right hereditary ring integral over its center has a hereditary center. Chatters and Jondrup [5] ask if a right and left hereditary PI ring has always has a hereditary center; we will produce an example to show it does not. -
Annihilation of Cohomology and Strong Generation of Module Categories,” International Mathematics Research Notices, Vol
Iyengar and Takahashi. (2015) “Annihilation of cohomology and strong generation of module categories,” International Mathematics Research Notices, Vol. 2015, Article ID IMRN-2014-599.R1, 21 pages. doi:10.1093/imrn/IMRN-2014-599.R1 Annihilation of cohomology and strong generation of module categories Srikanth B. Iyengar1 and Ryo Takahashi2 1Department of Mathematics, University of Utah, Salt Lake City, UT 84112-0090, USA and 2Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya 464-8602, Japan Correspondence to be sent to: [email protected] The cohomology annihilator of a noetherian ring that is finitely generated as a module over its center is introduced. Results are established linking the existence of non-trivial cohomology annihilators and the existence of strong generators for the category of finitely generated modules. Exploiting this link, results of Popescu and Roczen, and Wang concerning cohomology annihilators of commutative rings, and also results of Aihara and Takahashi, Keller and Van den Bergh, and Rouquier on strong finite generation of the corresponding bounded derived category, are generalized to cover excellent local rings and also rings essentially of finite type over a field. 1 Introduction The central theme of this article is that the two topics that make up its title are intimately related. Inklings of this can be found in the literature, both on annihilators of cohomology, notably work of Popescu and Roczen [26] from 1990, and on generators for module categories that is of more recent vintage; principally the articles of Dao and Takahashi [8], and Aihara and Takahashi [1]. We make precise the close link between the two topics, by introducing and developing appropriate notions and constructions, and use it to obtain more comprehensive results than are currently available in either one. -
Mittag-Leffler Modules and Semi-Hereditary Rings
RENDICONTI del SEMINARIO MATEMATICO della UNIVERSITÀ DI PADOVA ULRICH ALBRECHT ALBERTO FACCHINI Mittag-Leffler modules and semi-hereditary rings Rendiconti del Seminario Matematico della Università di Padova, tome 95 (1996), p. 175-188 <http://www.numdam.org/item?id=RSMUP_1996__95__175_0> © Rendiconti del Seminario Matematico della Università di Padova, 1996, tous droits réservés. L’accès aux archives de la revue « Rendiconti del Seminario Matematico della Università di Padova » (http://rendiconti.math.unipd.it/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Mittag-Leffler Modules And Semi-hereditary Rings. ULRICH ALBRECHT (*) - ALBERTO FACCHINI (**) (***) 1. - Introduction. In [2] it was demonstrated that many properties of torsion-free abelian groups carry over to non-singular right modules over right strongly non-singular, right semi-hereditary rings, where a ring R is called right strongly non-singular if the finitely generated non-singu- lar right modules are precisely the finitely generated submodules of free modules. A complete characterization of right strongly non-singu- lar right semi-hereditary rings can be found in [9, Theorem 5.18]. In particular, it was shown that right strongly non-singular, right semi- hereditary rings are left semi-hereditary too, so that we shall call such rings right strongly non-singular semi-hereditary. Examples of this type of rings are the semi-prime semi-hereditary right and left Goldie rings, for instance Prffer domains, as well as infinite dimensional rings like Z~’ .