DOCSLIB.ORG

A Class of Divisible Modules

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
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. ^7~ is called hereditary if it is closed under submodules. Modules in J7~ are called torsion. Every torsion class determines in every A e R^ a unique maximal torsion submodule ^~{A) is called the torsion submodule of A, and J7~{A\j?~{A)) = 0. Modules in ^~ = {Ae R^t | J7~(A) = 0} are called torsίonfree, and the torsionfree class Jf is closed under submodules, extensions, and direct products. ^ is hereditary if and only if j^~ is closed under injective envelopes. Throughout this paper, ^~ will denote a hereditary torsion class will denote the torsionfree class corresponding to j?~\ hence is a hereditary torsion theory [4], [10], and [17]. Each such is uniquely associated with a topologizing and idempotent filter F(^r) - {I \ JG R and R/IeJ^} of left ideals of R. In [5] the right derived functors for a hereditary torsion class J7~ are examined. These derived functors are given by - ^(A), E'jr (A) - 653 654 MARK L. TEPLY and R%(A) = Rn^\E{A)\A) for n ^ 2. If A is injective, then R^(A) = 0 for Λ ^ 1. If ie/; then BιAA) = (^~E(A)/A). A module M is called ^-injective ([8], [9], and [15]) if Ext1 (Γ, Af) = 0 for all Γey; (Note: in [14] ^injective modules are called i?7(^")-divisible.) This happens exactly when ^(E(M)/M) = 0 (see [8]). Hence a module A e ^ is ^injective if and only if RAA) = 0. If 0 > A >B >C >0 is any short exact sequence, then so is the induced sequence 0 JΓ(B) -> JT(C) -> B\r(A\ - RιAB) - RAQ -> R2AΛ) - . A torsion class ^~ is called stable if ^~ is closed under injective envelopes. R'AT) = 0 for all Te_^l and if ^" is stable, then RAT) = 0 for all Γe^ and for all n ^ 1. This and the long exact sequence in the preceding paragraph imply that if ^~ is stable and B e R^fί, then RAB) = 0 if and only if RAB/^~(B)) - 0. The submodule EAM) of jE'(ikf) is the (unique) largest module satisfying MS E^(M) S #(ikf) and EAM)/Me^. We call JS7^(Λf) the ^-injective envelope of Λf. E^(M) exists [14] and is .^injective. If iϋe^7 then we will use Q^- to denote EAR)* Q^ has a ring structure [8] and has received attention in a large number of papers (e.g.[7], [8], [9], and [15]). A module M is said to split if J7~(M) is a direct summand of M. We now state a lemma giving some properties of ^injective modules. LEMMA 1.1. (1) S~ is stable if and only if every ^-injective modules splits. (2) A direct summand of a Jf~-injective module is ^"-injective. (3) If A is a ^-injective module and if θ: A—>B is an epi- morphism, then θ~ι{^~(B)) = {x e A \ θ(x) e ^(B)} is ^-injeetive. (4) If j^Γ is stable, then every ^-injective module in j^7~ is infective. Proof. (1) If ^ is stable and A is ^injective, then E(A)/A so JT(E(A)) S A. By [2, Prop. 2.1], E(A) - ^~(E(A)) 0 F for some Fe^. Hence A = J^(#(A)) © (F n A). Since J^(#(A)) - ^~(A) in this case, then A splits. The converse is immediate from [2, Prop. 2.1]. (2) This is well-known and straight forward to prove. (3) Note that - A g(A) _ B A CLASS OF DIVISIBLE MODULES 655 So by the definition of ^injective envelope, E^(θ~ι(JΓ~(B))) = (4) Let Tej^~ be j^injective; so E(T)e^ and E(T)/TejK Since j?~ is closed under homomorphic images, E(T)/T e J7~ Π ^ = {0}; thus E{T) = Γ. We now turn our attention to an important special torsion theory. The class & of Goldie torsion modules is the smallest class contain- ing all isomorphic copies of all factor modules A/B, where B is an essential submodule of A (see [1], [2], [16], and [17]). The correspond- ing Goldie torsionfree class Λ^ is exactly the class of nonsingular modules. Sf is hereditary and stable; if R e <ylr, then & is precisely the class of singular modules. So if R is a commutative integral domain, then %? and <yίr coincide with the usual torsion and torsion- free classes, respectively. This makes (g^, Λ^) a "natural" torsion theory to consider when one is trying to generalize the usual results about torsion modules over an integral domain. Moreover, the &- injective modules are just the injective modules; and if R e ^V, then Q, ~ E(R) and the filter F(&) is the set of essential left ideals of R. 2* .^divisible modules* Historically the class of divisible modules has been very useful in studying modules over a commuta- tive integral domain. Frequently subclasses of the class of divisible modules have also yielded interesting results (e.g. see [11] and [12]). One such subclass studied by Matlis is the class of h-divisible modules, i.e., those modules which are homomorphic images of injective modules. A class related to the Λ-divisible modules is the class of divisible modules M such that every nonzero homomorphic image of M has a nonzero A-divisible submodule. For reasons indicated later, we refer to this latter class as the ^/-divisible modules. We shall show that not only does the class of ^-divisible modules satisfy many interest- ing properties but also it allows us to "smooth out" some of Matlis' results. For example, Matlis proves [11, Cor. 2.6] that every divisible module splits if and only if (1) h.d. Q ^ 1 (where Q is the quotient field of R) and (2) T = Ext1 (Q/R, T) for every torsion divisible module T with 0 as its only ^-divisible submodule. We remove condition (2) by only considering ^-divisible modules: every ^-divisible module splits if and only if h.d. Q <^ 1 (see Corollary 4.6). However, we will not limit our investigation to modules over a commutative integral domain. We are able to state our entire theory in terms of hereditary torsion theories (^, JF*) of modules over an associative ring R with identity element. To do this, we begin by defining subclasses ^a of R^fS for each ordinal number a. 656 MARK L. TEPLY ^ = {M e R^£ I M is a homomorphic image of a direct sum of ^^injective modules}. If a is not a limit ordinal, then <g^ = {Me ^ I IN S Λί" such that iVe ^_t and M/ΛΓe <gy. If a is a limit ordinal, then <g^ = {Λf e R^€ \ M = U JJ< JVjs where JV), 6 ^ for /S < a}. This transίinite definition is similar to the construction of a Loewy series [6]. Clearly <g^ S ^α for β ^a. It is straight forward to show that each <g^ is closed under homomorphic images and direct sums. If R is a commutative integral domain and if the torsion class is taken to be ^, it follows that each ^ is a subclass of the (usual) class of divisible modules. Also, over a commutative integral domain, any divisible module in ^ is injective. These two facts, plus the fact that each ^ depends on ^7 motivates the terminology, "^divisible module", for any hereditary torsion theory (_^7 ^) over anY ring. DEFINITION. A module M is called ^^divisible if and only if (i) M) is .^injective and (ii) ^~(M) e ^a for some ordinal a. will denote the class of all ^divisible modules. EXAMPLE 2.1. If R is a commutative integral domain, then a module M is ^-divisible if and only if (a) M is divisible and (b) every nonzero homomorphic image of M has a nonzero A-divisible submodule. Let MeQί^. From the construction of the classes ^, the closure properties of divisible modules, and the definition of gf-divisi- ble, it follows that M is divisible. Let θ: M—> X be an epimorphism. If ker θ does not contain gf (ikf), then it follows from (ii) that (&(M) + ker#)/ker θ contains an /^-divisible submodule.
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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; ·).
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Unique Factorization of Ideals in OK
    IDEAL FACTORIZATION KEITH CONRAD 1. Introduction We will prove here the fundamental theorem of ideal theory in number fields: every nonzero proper ideal in the integers of a number field admits unique factorization into a product of nonzero prime ideals. Then we will explore how far the techniques can be generalized to other domains. Definition 1.1. For ideals a and b in a commutative ring, write a j b if b = ac for an ideal c. Theorem 1.2. For elements α and β in a commutative ring, α j β as elements if and only if (α) j (β) as ideals. Proof. If α j β then β = αγ for some γ in the ring, so (β) = (αγ) = (α)(γ). Thus (α) j (β) as ideals. Conversely, if (α) j (β), write (β) = (α)c for an ideal c. Since (α)c = αc = fαc : c 2 cg and β 2 (β), β = αc for some c 2 c. Thus α j β in the ring. Theorem 1.2 says that passing from elements to the principal ideals they generate does not change divisibility relations. However, irreducibility can change. p Example 1.3. In Z[ −5],p 2 is irreduciblep as an element but the principal ideal (2) factors nontrivially: (2) = (2; 1 + −5)(2; 1 − −p5). p To see that neither of the idealsp (2; 1 + −5) and (2; 1 − −5) is the unit ideal, we give two arguments. Suppose (2; 1 + −5) = (1). Then we can write p p p 1 = 2(a + b −5) + (1 + −5)(c + d −5) for some integers a; b; c, and d.
    [Show full text]
  • Unique Factorization and Class Groups TCU Seminar Lecture Notes
    Unique Factorization and Class Groups TCU Seminar Lecture Notes George T. Gilbert Department of Mathematics, Texas Christian University [email protected] November 10, 17, & 29, 2011 1 Unique Factorization Our rings R will be integral domains (with identity). Definition. α1; α2 2 R are associates if α1 = α2u for some unit u 2 R. Definition. π 2 R is irreducible if π = π1π2 implies either π1 or π2 is a unit. Definition. R is a unique factorization domain (UFD) if every non-zero non-unit is a product of irreducible elements in a way that is unique up to order and associates. Examples: Z, any field k, the polynomial ring k[x1; : : : ; xk] GCDs exist in any UFD. (Greatest is in the sense of divisibility.) Theorem. If R is a UFD, then so is R[x]. n Proof. Let R be a UFD and K its field of fractions. The content of a0 + a1x + ··· + anx 2 K[x] is the GCD of a0; a1; : : : ; an. Primitive means content 1. Lemma. The content is multiplicative. Corollary. (Gauss' Lemma) The product of primitive polynomials is primitive. Lemma. A primitive polynomial is irreducible in R[x] if and only if it is irreducible in K[x]. Now use that K[x] is a UFD. Definition. R is a principal ideal domain (PID) if every ideal has the form (α) for some α 2 R. Theorem. In a PID R, a greatest common divisor g of α and β may be written as an R-linear combination of α and β. 1 Proof.
    [Show full text]
  • On Graded 2-Absorbing and Graded Weakly 2-Absorbing Pri- Mary Ideals
    KYUNGPOOK Math. J. 57(2017), 559-580 https://doi.org/10.5666/KMJ.2017.57.4.559 pISSN 1225-6951 eISSN 0454-8124 c Kyungpook Mathematical Journal On Graded 2-Absorbing and Graded Weakly 2-Absorbing Pri- mary Ideals Fatemeh Soheilnia∗ Department of Pure Mathematics, Faculty of Science, Imam Khomeini Interna- tional University, Qazvin, Iran e-mail : [email protected] Ahmad Yousefian Darani Department of Mathematics and Applications, University of Mohaghegh Ardabili, 5619911367, Ardabil, Iran e-mail : [email protected] Abstract. Let G be an arbitrary group with identity e and let R be a G-graded ring. In this paper, we define the concept of graded 2-absorbing and graded weakly 2-absorbing primary ideals of commutative G-graded rings with non-zero identity. A number of re- sults and basic properties of graded 2-absorbing primary and graded weakly 2-absorbing primary ideals are given. 1. Introduction Prime ideals (resp,. primary ideals) play an important role in commutative ring theory. Also, weakly prime ideals have been introduced and studied by D. D. Anderson and E. Smith in [1] and weakly primary ideals have been introduced and studied by S. E. Atani and F. Farzalipour in [2]. Later, A. Badawi in [6] generalized the concept of prime ideals in a different way. He defined a non-zero proper ideal I of commutative ring R to be a 2-absorbing ideal, if whenever a; b; c 2 R with abc 2 I, then either ab 2 I or ac 2 I or bc 2 I. This concept has a generalization that is called weakly 2-absorbing ideal, which has studied by A.
    [Show full text]
  • Monoid Rings and Strongly Two-Generated Ideals
    California State University, San Bernardino CSUSB ScholarWorks Electronic Theses, Projects, and Dissertations Office of aduateGr Studies 6-2014 MONOID RINGS AND STRONGLY TWO-GENERATED IDEALS Brittney M. Salt California State University - San Bernardino Follow this and additional works at: https://scholarworks.lib.csusb.edu/etd Part of the Algebra Commons, and the Other Mathematics Commons Recommended Citation Salt, Brittney M., "MONOID RINGS AND STRONGLY TWO-GENERATED IDEALS" (2014). Electronic Theses, Projects, and Dissertations. 31. https://scholarworks.lib.csusb.edu/etd/31 This Thesis is brought to you for free and open access by the Office of aduateGr Studies at CSUSB ScholarWorks. It has been accepted for inclusion in Electronic Theses, Projects, and Dissertations by an authorized administrator of CSUSB ScholarWorks. For more information, please contact [email protected]. Monoid Rings and Strongly Two-Generated Ideals A Thesis Presented to the Faculty of California State University, San Bernardino In Partial Fulfillment of the Requirements for the Degree Master of Arts in Mathematics by Brittney Salt June, 2014 Monoid Rings and Strongly Two-Generated Ideals A Thesis Presented to the Faculty of California State University, San Bernardino by Brittney Salt June, 2014 Approved by: J. Paul Vicknair, Committee Chair Date Zahid Hasan, Committee Member Jim Okon, Committee Member Peter Williams, Chair, Charles Stanton Department of Mathematics Graduate Coordinator, Department of Mathematics iii Abstract This paper determines whether monoid rings with the two-generator property have the strong two-generator property. Dedekind domains have both the two-generator and strong two-generator properties. How common is this? Two cases are considered here: the zero-dimensional case and the one-dimensional case for monoid rings.
    [Show full text]
  • 3 Properties of Dedekind Domains
    18.785 Number theory I Fall 2018 Lecture #3 09/12/2018 3 Properties of Dedekind domains In the previous lecture we defined a Dedekind domain as a noetherian domain A that satisfies either of the following equivalent conditions: • the localizations of A at its nonzero prime ideals are all discrete valuation rings; • A is integrally closed and has dimension at most one. In this lecture we will establish several additional properties enjoyed by Dedekind domains, the most significant of which is unique factorization of ideals. As we noted last time, Dedekind domains are typically not unique factorization domains (this occurs if and only if it is also a principal ideal domain), but ideals can be uniquely factored into prime ideals. 3.1 Invertible fractional ideals and the ideal class group In this section A is a noetherian domain (not necessarily a Dedekind domain) and K is its fraction field. Recall that a fractional ideal of A is a finitely generated A-submodule of K, and if I and J are fractional ideals, so is the colon ideal (I : J) := fx 2 K : xJ ⊆ Ig; and we that a fractional ideal I is invertible if IJ = A for some fractional ideal J. The definition of (A : I) implies I(A : I) ⊆ A, and Lemma 2.20 implies that I is invertible precisely when this inclusion is an equality, in which case the inverse of I is (A : I). Ideal multiplication is commutative and associative, thus the set of nonzero fractional ideals of a noetherian domain form an abelian monoid under multiplication with A = (1) as the identity.
    [Show full text]