Other-Half-Factoriality in Commutative Monoids and Integral Domains
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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. -
Exercises and Solutions in Groups Rings and Fields
EXERCISES AND SOLUTIONS IN GROUPS RINGS AND FIELDS Mahmut Kuzucuo˘glu Middle East Technical University [email protected] Ankara, TURKEY April 18, 2012 ii iii TABLE OF CONTENTS CHAPTERS 0. PREFACE . v 1. SETS, INTEGERS, FUNCTIONS . 1 2. GROUPS . 4 3. RINGS . .55 4. FIELDS . 77 5. INDEX . 100 iv v Preface These notes are prepared in 1991 when we gave the abstract al- gebra course. Our intention was to help the students by giving them some exercises and get them familiar with some solutions. Some of the solutions here are very short and in the form of a hint. I would like to thank B¨ulent B¨uy¨ukbozkırlı for his help during the preparation of these notes. I would like to thank also Prof. Ismail_ S¸. G¨ulo˘glufor checking some of the solutions. Of course the remaining errors belongs to me. If you find any errors, I should be grateful to hear from you. Finally I would like to thank Aynur Bora and G¨uldaneG¨um¨u¸sfor their typing the manuscript in LATEX. Mahmut Kuzucuo˘glu I would like to thank our graduate students Tu˘gbaAslan, B¨u¸sra C¸ınar, Fuat Erdem and Irfan_ Kadık¨oyl¨ufor reading the old version and pointing out some misprints. With their encouragement I have made the changes in the shape, namely I put the answers right after the questions. 20, December 2011 vi M. Kuzucuo˘glu 1. SETS, INTEGERS, FUNCTIONS 1.1. If A is a finite set having n elements, prove that A has exactly 2n distinct subsets. -
The Quotient Field of an Intersection of Integral Domains
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 70, 238-249 (1981) The Quotient Field of an Intersection of Integral Domains ROBERT GILMER* Department of Mathematics, Florida State University, Tallahassee, Florida 32306 AND WILLIAM HEINZER' Department of Mathematics, Purdue University, West Lafayette, Indiana 47907 Communicated by J. DieudonnP Received August 10, 1980 INTRODUCTION If K is a field and 9 = {DA} is a family of integral domains with quotient field K, then it is known that the family 9 may possess the following “bad” properties with respect to intersection: (1) There may exist a finite subset {Di}rzl of 9 such that nl= I Di does not have quotient field K; (2) for some D, in 9 and some subfield E of K, the quotient field of D, n E may not be E. In this paper we examine more closely conditions under which (1) or (2) occurs. In Section 1, we work in the following setting: we fix a subfield F of K and an integral domain J with quotient field F, and consider the family 9 of K-overrings’ of J with quotient field K. We then ask for conditions under which 9 is closed under intersection, or finite intersection, or under which D n E has quotient field E for each D in 9 and each subfield E of K * The first author received support from NSF Grant MCS 7903123 during the writing of this paper. ‘The second author received partial support from NSF Grant MCS 7800798 during the writing of this paper. -
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; ·). -
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. -
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. -
A Class of Divisible Modules
Pacific Journal of Mathematics A CLASS OF DIVISIBLE MODULES MARK LAWRENCE TEPLY Vol. 45, No. 2 October 1973 PACIFIC JOURNAL OF MATHEMATICS Vol. 45, No. 2, 1973 A CLASS OF DIVISIBLE MODULES MARK L. TEPLY The ^^divisible jβ-modules are defined in terms of a hereditary torsion theory of modules over an associative ring R with identity element. In the special case where J^~ is the usual torsion class of modules over a commutative in- tegral domain, the class of J^divisible modules is precisely the class of divisible modules M such that every nonzero homomorphic image of M has a nonzero /^-divisible submodule. In general, if J7~ is a stable hereditary torsion class, the class of ^-divisible modules satisfies many of the traditional properties of divisible modules over a commutative integral domain. This is especially true when J^~ is Goldie's torsion class 2^. For suitable ^\ the splitting of all ^divisible modules is equivalent to h.d.Q^- ^ 1, where Q^ is the ring of quotients naturally associated with ^T Generalizations of Dedekind domains are studied in terms of ^^divisibility. 1* Notation, terminology, and preliminary results* In this paper, all rings R are associative rings with identity element, and all modules are unitary left iϋ-modules. B^f denotes the category of all left i?-modules. E(M) denotes the injective envelope of Me R<^fί. In homological expressions, the "R" will be omitted for convenience 1 in printing (e.g. Ext^ = Ext and h.d RQ — h.d. Q). Following S.E. Dickson [4], we call a nonempty subclass J7~ of R^/^ a torsion class if ^ is closed under factors, extensions, and arbitrary direct sums. -
Dedekind Domains and Rings of Quotients
Pacific Journal of Mathematics DEDEKIND DOMAINS AND RINGS OF QUOTIENTS LUTHER ELIC CLABORN Vol. 15, No. 1 September 1965 PACIFIC JOURNAL OF MATHEMATICS Vol. 15, No. 1. 1965 DEDEKIND DOMAINS AND RINGS OF QUOTIENTS LUTHER CLABORN We study the relation of the ideal class group of a Dedekind domain A to that of As, where S is a multiplicatively closed subset of A. We construct examples of (a) a Dedekind domain with no principal prime ideal and (b) a Dedekind domain which is not the integral closure of a principal ideal domain. We also obtain some qualitative information on the number of non-principal prime ideals in an arbitrary Dedekind domain. If A is a Dadekind domain, S the set of all monic poly- nomials and T the set of all primitive polynomials of A[X], then A[X]<? and A[X]T are both Dadekind domains. We obtain the class groups of these new Dsdekind domains in terms of that of A. 1* LEMMA 1-1. If A is a Dedekind domain and S is a multi- plicatively closed set of A suoh that As is not a field, then As is also a Dedekind domain. Proof. That As is integrally closed and Noetherian if A is, follows from the general theory of quotient ring formations. The primes of As are of the type PAS) where P is a prime ideal of A such that PΓ)S = ψ. Since height PAS = height P if PΠS = φ, P Φ (0) and PΠS = φ imply that height PAS = 1. PROPOSITION 1-2. -
Free Medial Quandles
Algebra Univers. 78 (2017) 43–54 DOI 10.1007/s00012-017-0443-2 Published online May 23, 2017 © 2017 The Author(s) Algebra Universalis This article is an open access publication Free medial quandles Premyslˇ Jedlicka,ˇ Agata Pilitowska, and Anna Zamojska-Dzienio Abstract. This paper gives the construction of free medial quandles as well as free n-symmetric medial quandles and free m-reductive medial quandles. 1. Introduction A binary algebra (Q, ) is called a rack if the following conditions hold, for · every x, y, z Q: ∈ x(yz)=(xy)(xz) (we say Q is left distributive), • the equation xu = y has a unique solution u Q (we say Q is a left • ∈ quasigroup). An idempotent rack is called a quandle (we say Q is idempotent if xx = x for every x Q). A quandle Q is medial if, for every x, y, u, v Q, ∈ ∈ (xy)(uv)=(xu)(yv). An important example of a medial quandle is an abelian group A with an operation defined by a b = (1 h)(a)+h(b), where h is an automorphism ∗ ∗ − of A. This construction is called an affine quandle (or sometimes an Alexander quandle) and denoted by Aff(A, h). In the literature [6, 7], the group A is 1 usually considered to be a Z[t, t− ]-module, where t a = h(a), for each a A. · ∈ We shall adopt this point of view here as well and we usually write Aff(A, r) instead, where r is a ring element. Note that in universal algebra terminology, an algebra is said to be affine if it is polynomially equivalent to a module. -
Clifford Regular Domains1
Journal of Algebra 238, 703᎐722Ž. 2001 doi:10.1006rjabr.2000.8670, available online at http:rrwww.idealibrary.com on Clifford Regular Domains1 S. Bazzoni Dipartimento di Matematica Pura e Applicata, Uni¨ersita` di Pado¨a, Via Belzoni 7, View metadata, citation and similar papers at core.ac.uk35131 Pado¨a, Italy brought to you by CORE E-mail: [email protected] provided by Elsevier - Publisher Connector Communicated by Kent R. Fuller Received May 8, 2000 The class semigroup of a commutative integral domain R is the semigroup S Ž.R of the isomorphism classes of the nonzero ideals of R with operation induced by multiplication. A domain R is said to be Clifford regular if S Ž.R is a Clifford semigroup, i.e. S Ž.R is the disjoint union of the subgroups associated to the idempotent elements. In this paper we characterize the noetherian and the integrally closed Clifford regular domains and find some properties of an arbitrary Clifford regular domain. ᮊ 2001 Academic Press Key Words: Clifford semigroup; stable domains; Prufer¨ domains; finite character. INTRODUCTION The notion of the class group of a commutative domain is a classical one and it is one of the major objects of investigation in commutative algebra. The class group CŽ.R of a domain R consists of the isomorphism classes of the invertible ideals of R. We generalize this notion by considering the class semigroup S Ž.R of R consisting of the isomorphism classes of all the nonzero ideals of R. Throughout, R will denote a commutative domain. -
Part IV (§18-24) Rings and Fields
Part IV (§18-24) Rings and Fields Satya Mandal University of Kansas, Lawrence KS 66045 USA January 22 18 Rings and Fields Groups dealt with only one binary operation. You are used to working with two binary operation, in the usual objects you work with: Z, R. Now we will study with such objects, with two binary operations: additions and multiplication. 18.1 Defintions and Basic Properties Definition 18.1. A Ring hR, +, ·i is a set with two binary operations +, ·, which we call addition and multiplication, defined on R such that 1. hR, +, ·i is an abelian group. The additive identiy is denoted by zero 0. 2. Multiplication is associative. 3. The distributive property is satiesfied as follows: ∀ a,b,c ∈ R a(b + c)= ab + ac and (b + c)a = ba + ca. 1 Further, A ring hR, +, ·i is called a commutative ring, if the multipli- cation is commutative. That means, if ∀ a,b ∈ R ab = ba. Definition 18.2. Let hR, +, ·i be a ring. If R has a multiplicative identity, then we has hR, +, ·i is a ring with unity. The multiplicative unit is denoted by 1. Recall, it means ∀ x ∈ R 1 · x = x · 1= x. Barring some exceptions (if any), we consider rings with unity only. Lemma 18.3. Suppose hR, +, ·i is a ring with unity. Suppse 0=1. then R = {0}. Proof. Suppose x ∈ R. Then, x = x · 1= x · 0= x · (0+0) = x · 0+ x · 0= x · 1+ x · 1= x + x. So, x + x = x and hence x = 0. -
The Lattice of Quasivarietes of Modules Over a Dedekind Ring∗
“adm-n1” — 2019/3/22 — 12:03 — page 37 — #45 Algebra and Discrete Mathematics RESEARCH ARTICLE Volume 27 (2019). Number 1, pp. 37–49 c Journal “Algebra and Discrete Mathematics” The lattice of quasivarietes of modules over a Dedekind ring∗ Přemysl Jedlička, Katarzyna Matczak, Anna Mućka Communicated by A. I. Kashu Abstract. In 1995 D. V. Belkin described the lattice of qua- sivarieties of modules over principal ideal domains [1]. The following paper provides a description of the lattice of subquasivarieties of the variety of modules over a given Dedekind ring. It also shows which subvarieties of these modules are deductive (a variety is deductive if every subquasivariety is a variety). Introduction There are many algebraic structures that are polynomially equivalent to modules; typical examples are abelian algebras with a Mal’tsev term. However, in most cases these modules are over rings with no interesting algebraic properties. There are exceptions though, for instance finite idempotent entropic quasigroups. Since the work of Toyoda [12], we know that a finite idempotent entropic quasigroup is polynomially equivalent to a module over Z[x]/(xn 1), for some n, and these rings are well described − in commutative algebra; we know, for instance, that they are products of pairwise different Dedekind domains [14]. Since the polynomial equivalence preserves congruences, it is natural to study varieties and quasivarieties in better understood equivalent algebras, ∗ The joint research within the framework of the Polish-Czech cooperation grant no. 7AMB13PL013 and no. 8829/R13/R14. 2010 MSC: 08A62, 08C15, 20N02, 20N05. Key words and phrases: quasivarieties, lattices, modules, Dedekind rings.