DOCSLIB.ORG

Localization and Classification of Zero-Divisor Graphs of Rings

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Localization and Classification of Zero-Divisor Graphs of Rings Localization and Classification of Zero-divisor Graphs of Rings Tom Cuchta, Kathryn Lokken, William Young Marshall University, University of Wisconsin, Purdue University Localization and Classification of Zero-divisor Graphs of Rings – p. 1/37 Basic ring theory • A ring R is a set with two operations: + and ·, and it has most of the nice properties that you "would like" an algebraic structure to have. Localization and Classification of Zero-divisor Graphs of Rings – p. 2/37 Basic ring theory • A ring R is a set with two operations: + and ·, and it has most of the nice properties that you "would like" an algebraic structure to have. • In general, rings do not have the property of commutativity on multiplication or a multiplicative identity, but for this talk, all rings have both. Localization and Classification of Zero-divisor Graphs of Rings – p. 2/37 Basic ring theory • A ring R is a set with two operations: + and ·, and it has most of the nice properties that you "would like" an algebraic structure to have. • In general, rings do not have the property of commutativity on multiplication or a multiplicative identity, but for this talk, all rings have both. • Such rings are called commutative rings with identity. Localization and Classification of Zero-divisor Graphs of Rings – p. 2/37 Basic ring theory The best way to learn about rings is to look at some: • The integers, Z, notice no elements have multiplicative inverses other than 1 and −1 Localization and Classification of Zero-divisor Graphs of Rings – p. 3/37 Basic ring theory The best way to learn about rings is to look at some: • The integers, Z, notice no elements have multiplicative inverses other than 1 and −1 • The integers modulo n, Zn, is a ring under addition modulo n and multiplication modulo n Localization and Classification of Zero-divisor Graphs of Rings – p. 3/37 Basic ring theory The best way to learn about rings is to look at some: • The integers, Z, notice no elements have multiplicative inverses other than 1 and −1 • The integers modulo n, Zn, is a ring under addition modulo n and multiplication modulo n • The 2 × 2 matrices with real entries Localization and Classification of Zero-divisor Graphs of Rings – p. 3/37 Basic ring theory The best way to learn about rings is to look at some: • The integers, Z, notice no elements have multiplicative inverses other than 1 and −1 • The integers modulo n, Zn, is a ring under addition modulo n and multiplication modulo n • The 2 × 2 matrices with real entries • Z2[x], polynomials with coefficients from Z2 Localization and Classification of Zero-divisor Graphs of Rings – p. 3/37 Basic ring theory • An element z ∈ R is called a zero-divisor if there is a nonzero z′ ∈ R such that zz′ = 0. The set of zero-divisors is denoted Z(R). Localization and Classification of Zero-divisor Graphs of Rings – p. 4/37 Basic ring theory • An element z ∈ R is called a zero-divisor if there is a nonzero z′ ∈ R such that zz′ = 0. The set of zero-divisors is denoted Z(R). • For example, in Z12, 8 · 3 (mod 12) = 24 (mod 12) ≡ 0 (mod 12), so 8 and 3 are zero-divisors in Z12. Localization and Classification of Zero-divisor Graphs of Rings – p. 4/37 Basic ring theory • An element z ∈ R is called a zero-divisor if there is a nonzero z′ ∈ R such that zz′ = 0. The set of zero-divisors is denoted Z(R). • For example, in Z12, 8 · 3 (mod 12) = 24 (mod 12) ≡ 0 (mod 12), so 8 and 3 are zero-divisors in Z12. • An element u ∈ R is called a unit if there exists an element u−1 ∈ R such that uu−1 = 1. The set of units in R is denoted U(R). Localization and Classification of Zero-divisor Graphs of Rings – p. 4/37 Basic ring theory • A subring of a ring R is a subset of R that is also ring. Localization and Classification of Zero-divisor Graphs of Rings – p. 5/37 Basic ring theory • A subring of a ring R is a subset of R that is also ring. • An ideal, I, of R is an especially nice subring of R such that for every r ∈ R and i ∈ I, ri ∈ I. In a sense, an ideal is a subring that "absorbs" ring multiplication. Localization and Classification of Zero-divisor Graphs of Rings – p. 5/37 Basic ring theory • A subring of a ring R is a subset of R that is also ring. • An ideal, I, of R is an especially nice subring of R such that for every r ∈ R and i ∈ I, ri ∈ I. In a sense, an ideal is a subring that "absorbs" ring multiplication. • A prime ideal, P of R is an ideal of R with the following property: if r, r′ ∈ R such that rr′ ∈ P , then r ∈ P or r′ ∈ P . Localization and Classification of Zero-divisor Graphs of Rings – p. 5/37 Basic ring theory • If a ∈ R, then we define (a) to be the principal ideal generated by a. This is, specifically, the set of multiples of a. Localization and Classification of Zero-divisor Graphs of Rings – p. 6/37 Basic ring theory • If a ∈ R, then we define (a) to be the principal ideal generated by a. This is, specifically, the set of multiples of a. • For example, in Z12, (2) = {0, 2, 4, 6, 8, 10}. Localization and Classification of Zero-divisor Graphs of Rings – p. 6/37 Basic ring theory • If a ∈ R, then we define (a) to be the principal ideal generated by a. This is, specifically, the set of multiples of a. • For example, in Z12, (2) = {0, 2, 4, 6, 8, 10}. • A ring such that all ideals are principal is called a principal ideal ring. Localization and Classification of Zero-divisor Graphs of Rings – p. 6/37 Basic ring theory • If a ∈ R, then we define (a) to be the principal ideal generated by a. This is, specifically, the set of multiples of a. • For example, in Z12, (2) = {0, 2, 4, 6, 8, 10}. • A ring such that all ideals are principal is called a principal ideal ring. • For all n, Zn is a principal ideal ring. The prime ideals are generated by the prime numbers in this ring. Localization and Classification of Zero-divisor Graphs of Rings – p. 6/37 Basic ring theory • The direct product of two rings R1 and R2, denoted R1 × R2, is the set of ordered pairs (r1, r2) such that r1 ∈ R1 and r2 ∈ R2. Localization and Classification of Zero-divisor Graphs of Rings – p. 7/37 Basic ring theory • The direct product of two rings R1 and R2, denoted R1 × R2, is the set of ordered pairs (r1, r2) such that r1 ∈ R1 and r2 ∈ R2. • For example, Z2 × Z2 = {(0, 0), (1, 0), (0, 1), (1, 1)}. Localization and Classification of Zero-divisor Graphs of Rings – p. 7/37 Basic ring theory • The direct product of two rings R1 and R2, denoted R1 × R2, is the set of ordered pairs (r1, r2) such that r1 ∈ R1 and r2 ∈ R2. • For example, Z2 × Z2 = {(0, 0), (1, 0), (0, 1), (1, 1)}. • Operations in direct product rings are componentwise, meaning that (r1, r2) + (r3, r4) = (r1 + r3, r2 + r4) and (r1, r2)(r3, r4) = (r1r3, r2r4). Localization and Classification of Zero-divisor Graphs of Rings – p. 7/37 Basic ring theory • Two rings, R1 and R2, are said to be isomorphic if there exists a one-to-one and ′ onto map, φ : R1 −→ R2 such that if r, r ∈ R1, then φ(r + r′)= φ(r)+ φ(r′) and φ(rr′)= φ(r)φ(r′). Localization and Classification of Zero-divisor Graphs of Rings – p. 8/37 Basic ring theory • Two rings, R1 and R2, are said to be isomorphic if there exists a one-to-one and ′ onto map, φ : R1 −→ R2 such that if r, r ∈ R1, then φ(r + r′)= φ(r)+ φ(r′) and φ(rr′)= φ(r)φ(r′). • For example, there is an isomorphism between the rings Z6 and Z2 × Z3 through the function φ(0) = (0, 0),φ(1) = (1, 1),φ(2) = (0, 2),φ(3) = (1, 0),φ(4) = (0, 1), and φ(5) = (1, 2). Localization and Classification of Zero-divisor Graphs of Rings – p. 8/37 Localizations • Since we know that not all elements in rings have multiplicative inverses, we might want to extend a ring to have multiplicative inverses for some subset. The extended ring is called a localization, and it is comprised of r equivalence classes of fractions, denoted [s]. Localization and Classification of Zero-divisor Graphs of Rings – p. 9/37 Localizations • Since we know that not all elements in rings have multiplicative inverses, we might want to extend a ring to have multiplicative inverses for some subset. The extended ring is called a localization, and it is comprised of r equivalence classes of fractions, denoted [s]. • We will learn that, in fact, you have probably done this before! But first, some more definitions. Localization and Classification of Zero-divisor Graphs of Rings – p. 9/37 Localizations • Since we are dealing with fractions and their equivalence classes, a good place to start is to define the equivalence class we are using. Localization and Classification of Zero-divisor Graphs of Rings – p. 10/37 Localizations • Since we are dealing with fractions and their equivalence classes, a good place to start is to define the equivalence class we are using.
Load more

Recommended publications
  • Adjacency Matrices of Zero-Divisor Graphs of Integers Modulo N Matthew Young
    inv lve a journal of mathematics Adjacency matrices of zero-divisor graphs of integers modulo n Matthew Young msp 2015 vol. 8, no. 5 INVOLVE 8:5 (2015) msp dx.doi.org/10.2140/involve.2015.8.753 Adjacency matrices of zero-divisor graphs of integers modulo n Matthew Young (Communicated by Kenneth S. Berenhaut) We study adjacency matrices of zero-divisor graphs of Zn for various n. We find their determinant and rank for all n, develop a method for finding nonzero eigen- values, and use it to find all eigenvalues for the case n p3, where p is a prime D number. We also find upper and lower bounds for the largest eigenvalue for all n. 1. Introduction Let R be a commutative ring with a unity. The notion of a zero-divisor graph of R was pioneered by Beck[1988]. It was later modified by Anderson and Livingston [1999] to be the following. Definition 1.1. The zero-divisor graph .R/ of the ring R is a graph with the set of vertices V .R/ being the set of zero-divisors of R and edges connecting two vertices x; y R if and only if x y 0. 2 D To each (finite) graph , one can associate the adjacency matrix A./ that is a square V ./ V ./ matrix with entries aij 1, if vi is connected with vj , and j j j j D zero otherwise. In this paper we study the adjacency matrices of zero-divisor graphs n .Zn/ of rings Zn of integers modulo n, where n is not prime.
    [Show full text]
  • Zero Divisor Conjecture for Group Semifields
    Ultra Scientist Vol. 27(3)B, 209-213 (2015). Zero Divisor Conjecture for Group Semifields W.B. VASANTHA KANDASAMY1 1Department of Mathematics, Indian Institute of Technology, Chennai- 600 036 (India) E-mail: [email protected] (Acceptance Date 16th December, 2015) Abstract In this paper the author for the first time has studied the zero divisor conjecture for the group semifields; that is groups over semirings. It is proved that whatever group is taken the group semifield has no zero divisors that is the group semifield is a semifield or a semidivision ring. However the group semiring can have only idempotents and no units. This is in contrast with the group rings for group rings have units and if the group ring has idempotents then it has zero divisors. Key words: Group semirings, group rings semifield, semidivision ring. 1. Introduction study. Here just we recall the zero divisor For notions of group semirings refer8-11. conjecture for group rings and the conditions For more about group rings refer 1-6. under which group rings have zero divisors. Throughout this paper the semirings used are Here rings are assumed to be only fields and semifields. not any ring with zero divisors. It is well known1,2,3 the group ring KG has zero divisors if G is a 2. Zero divisors in group semifields : finite group. Further if G is a torsion free non abelian group and K any field, it is not known In this section zero divisors in group whether KG has zero divisors. In view of the semifields SG of any group G over the 1 problem 28 of .
    [Show full text]
  • On Finite Semifields with a Weak Nucleus and a Designed
    On finite semifields with a weak nucleus and a designed automorphism group A. Pi~nera-Nicol´as∗ I.F. R´uay Abstract Finite nonassociative division algebras S (i.e., finite semifields) with a weak nucleus N ⊆ S as defined by D.E. Knuth in [10] (i.e., satisfying the conditions (ab)c − a(bc) = (ac)b − a(cb) = c(ab) − (ca)b = 0, for all a; b 2 N; c 2 S) and a prefixed automorphism group are considered. In particular, a construction of semifields of order 64 with weak nucleus F4 and a cyclic automorphism group C5 is introduced. This construction is extended to obtain sporadic finite semifields of orders 256 and 512. Keywords: Finite Semifield; Projective planes; Automorphism Group AMS classification: 12K10, 51E35 1 Introduction A finite semifield S is a finite nonassociative division algebra. These rings play an important role in the context of finite geometries since they coordinatize projective semifield planes [7]. They also have applications on coding theory [4, 8, 6], combinatorics and graph theory [13]. During the last few years, computational efforts in order to clasify some of these objects have been made. For instance, the classification of semifields with 64 elements is completely known, [16], as well as those with 243 elements, [17]. However, many questions are open yet: there is not a classification of semifields with 128 or 256 elements, despite of the powerful computers available nowadays. The knowledge of the structure of these concrete semifields can inspire new general constructions, as suggested in [9]. In this sense, the work of Lavrauw and Sheekey [11] gives examples of constructions of semifields which are neither twisted fields nor two-dimensional over a nucleus starting from the classification of semifields with 64 elements.
    [Show full text]
  • 6. Localization
    52 Andreas Gathmann 6. Localization Localization is a very powerful technique in commutative algebra that often allows to reduce ques- tions on rings and modules to a union of smaller “local” problems. It can easily be motivated both from an algebraic and a geometric point of view, so let us start by explaining the idea behind it in these two settings. Remark 6.1 (Motivation for localization). (a) Algebraic motivation: Let R be a ring which is not a field, i. e. in which not all non-zero elements are units. The algebraic idea of localization is then to make more (or even all) non-zero elements invertible by introducing fractions, in the same way as one passes from the integers Z to the rational numbers Q. Let us have a more precise look at this particular example: in order to construct the rational numbers from the integers we start with R = Z, and let S = Znf0g be the subset of the elements of R that we would like to become invertible. On the set R×S we then consider the equivalence relation (a;s) ∼ (a0;s0) , as0 − a0s = 0 a and denote the equivalence class of a pair (a;s) by s . The set of these “fractions” is then obviously Q, and we can define addition and multiplication on it in the expected way by a a0 as0+a0s a a0 aa0 s + s0 := ss0 and s · s0 := ss0 . (b) Geometric motivation: Now let R = A(X) be the ring of polynomial functions on a variety X. In the same way as in (a) we can ask if it makes sense to consider fractions of such polynomials, i.
    [Show full text]
  • Rings of Quotients and Localization a Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Ar
    RINGS OF QUOTIENTS AND LOCALIZATION A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREME N TS FOR THE DEGREE OF MASTER OF ARTS IN MATHEMATICS IN THE GRADUATE SCHOOL OF THE TEXAS WOMAN'S UNIVERSITY COLLEGE OF ARTS AND SCIENCE S BY LILLIAN WANDA WRIGHT, B. A. DENTON, TEXAS MAY, l974 TABLE OF CONTENTS INTRODUCTION . J Chapter I. PRIME IDEALS AND MULTIPLICATIVE SETS 4 II. RINGS OF QUOTIENTS . e III. CLASSICAL RINGS OF QUOTIENTS •• 25 IV. PROPERTIES PRESERVED UNDER LOCALIZATION 30 BIBLIOGRAPHY • • • • • • • • • • • • • • • • ••••••• 36 iii INTRODUCTION The concept of a ring of quotients was apparently first introduced in 192 7 by a German m a thematician Heinrich Grell i n his paper "Bezeihungen zwischen !deale verschievener Ringe" [ 7 ] . I n his work Grelt observed that it is possible to associate a ring of quotients with the set S of non -zero divisors in a r ing. The elements of this ring of quotients a r e fractions whose denominators b elong to Sand whose numerators belong to the commutative ring. Grell's ring of quotients is now called the classical ring of quotients. 1 GrelL 1 s concept of a ring of quotients remained virtually unchanged until 1944 when the Frenchman Claude C hevalley presented his paper, "On the notion oi the Ring of Quotients of a Prime Ideal" [ 5 J. C hevalley extended Gre ll's notion to the case wher e Sis the compte- ment of a primt: ideal. (Note that the set of all non - zer o divisors and the set- theoretic c om?lement of a prime ideal are both instances of .multiplicative sets -- sets tha t are closed under multiplicatio n.) 1According to V.
    [Show full text]
  • Mathematics 411 9 December 2005 Final Exam Preview
    Mathematics 411 9 December 2005 Final exam preview Instructions: As always in this course, clarity of exposition is as important as correctness of mathematics. 1. Recall that a Gaussian integer is a complex number of the form a + bi where a and b are integers. The Gaussian integers form a ring Z[i], in which the number 3 has the following special property: Given any two Gaussian integers r and s, if 3 divides the product rs, then 3 divides either r or s (or both). Use this fact to prove the following theorem: For any integer n ≥ 1, given n Gaussian integers r1, r2,..., rn, if 3 divides the product r1 ···rn, then 3 divides at least one of the factors ri. 2. (a) Use the Euclidean algorithm to find an integer solution to the equation 7x + 37y = 1. (b) Explain how to use the solution of part (a) to find a solution to the congruence 7x ≡ 1 (mod 37). (c) Use part (b) to explain why [7] is a unit in the ring Z37. 3. Given a Gaussian integer t that is neither zero nor a unit in Z[i], a factorization t = rs in Z[i] is nontrivial if neither r nor s is a unit. (a) List the units in Z[i], indicating for each one what its multiplicative inverse is. No proof is necessary. (b) Provide a nontrivial factorization of 53 in Z[i]. Explain why your factorization is nontriv- ial. (c) Suppose that p is a prime number in Z that has no nontrivial factorizations in Z[i].
    [Show full text]
  • Chapter 10 an Introduction to Rings
    Chapter 10 An Introduction to Rings 10.1 Definitions and Examples Recall that a group is a set together with a single binary operation, which together satisfy a few modest properties. Loosely speaking, a ring is a set together with two binary oper- ations (called addition and multiplication) that are related via a distributive property. Definition 10.1. A ring R is a set together with two binary operations + and (called addition and multiplication, respectively) satisfying the following: · (i) (R,+) is an abelian group. (ii) is associative: (a b) c = a (b c) for all a,b,c R. · · · · · 2 (iii) The distributive property holds: a (b + c)=(a b)+(a c) and (a + b) c =(a c)+(b c) · · · · · · for all a,b,c R. 2 Remark 10.2. We make a couple comments about notation. (a) We often write ab in place a b. · (b) The additive inverse of the ring element a R is denoted a. 2 − Theorem 10.3. Let R be a ring. Then for all a,b R: 2 1. 0a = a0=0 2. ( a)b = a( b)= (ab) − − − 3. ( a)( b)=ab − − Definition 10.4. A ring R is called commutative if multiplication is commutative. Definition 10.5. A ring R is said to have an identity (or called a ring with 1) if there is an element 1 R such that 1a = a1=a for all a R. 2 2 Exercise 10.6. Justify that Z is a commutative ring with 1 under the usual operations of addition and multiplication. Which elements have multiplicative inverses in Z? CHAPTER 10.
    [Show full text]
  • Student Notes
    270 Chapter 5 Rings, Integral Domains, and Fields 270 Chapter 5 Rings, Integral Domains, and Fields 5.2 Integral5.2 DomainsIntegral Domains and Fields and Fields In the precedingIn thesection preceding we definedsection we the defined terms the ring terms with ring unity, with unity, commutative commutative ring, ring,andandzerozero 270 Chapter 5 Rings, Integral Domains, and Fields 5.2divisors. IntegralAll Domains threedivisors. of and theseAll Fields three terms ! of these are usedterms inare defining used in defining an integral an integral domain. domain. Week 13 Definition 5.14DefinitionI Integral 5.14 IDomainIntegral Domain 5.2 LetIntegralD be a ring. Domains Then D is an andintegral Fields domain provided these conditions hold: Let D be a ring. Then D is an integral domain provided these conditions hold: In1. theD ispreceding a commutative section ring. we defined the terms ring with unity, commutative ring, and zero 1. D is a commutativedivisors.2. D hasAll a ring. unity three eof, and thesee 2terms0. are used in defining an integral domain. 2. D has a unity3. De,has and noe zero2 0 divisors.. Definition 5.14 I Integral Domain 3. D has no zero divisors. Note that the requirement e 2 0 means that an integral domain must have at least two Let D be a ring. Then D is an integral domain provided these conditions hold: Remark. elements. Note that the1. requirementD is a commutative e 2 0ring. means that an integral domain must have at least two elements. Example2. D has a1 unityThe e, ring andZe of2 all0. integers is an integral domain, but the ring E of all even in- tegers3.
    [Show full text]
  • FUSIBLE RINGS 1. Introduction It Is Well-Known That the Sum of Two Zero
    FUSIBLE RINGS EBRAHIM GHASHGHAEI AND WARREN WM. MCGOVERN∗ Abstract. A ring R is called left fusible if every nonzero element is the sum of a left zero-divisor and a non-left zero-divisor. It is shown that if R is a left fusible ring and σ is a ring automorphism of R, then R[x; σ] and R[[x; σ]] are left fusible. It is proved that if R is a left fusible ring, then Mn(R) is a left fusible ring. Examples of fusible rings are complemented rings, special almost clean rings, and commutative Jacobson semi-simple clean rings. A ring R is called left unit fusible if every nonzero element of R can be written as the sum of a unit and a left zero-divisor in R. Full Rings of continuous functions are fusible. It is also shown that if 1 = e1 + e2 + ::: + en in a ring R where the ei are orthogonal idempotents and each eiRei is left unit fusible, then R is left unit fusible. Finally, we give some properties of fusible rings. Keywords: Fusible rings, unit fusible rings, zero-divisors, clean rings, C(X). MSC(2010): Primary:16U99; Secondary:16W99, 13A99. 1. Introduction It is well-known that the sum of two zero-divisors need not be a zero-divisor. In [18, Theorem 1.12] the authors characterized commutative rings for which the set of zero-divisors is an ideal. The set of zero-divisors (i.e., Z(R)) is an ideal in a commutative ring R if and only if the classical ring of quotients of R (i.e., qcl(R)) is a local ring if and only if Z(R) is a prime ideal of R and qcl(R) is the localization at Z(R).
    [Show full text]
  • MODEL ANSWERS to HWK #7 1. Suppose That F Is a Field and That A
    MODEL ANSWERS TO HWK #7 1. Suppose that F is a field and that a and b are in F . Suppose that a · b = 0; and that b 6= 0. Let c be the inverse of b. Multiplying the equation above by c on the left, we get 0 = 0 · c = (a · b) · c = a · (b · c) = a · 1 = a: Thus a = 0. It follows that F is an integral domain. 2. We compute a b 1 0 a 0 = c d 0 0 c 0 and 1 0 a b a b = : 0 0 c d 0 0 If we have equality, then b = 0 and c = 0 and vice-versa. Thus the only matrices that commute with 1 0 0 0 are matrices of the form a 0 ; 0 b where a and b are arbitrary. 3. (a) One way to do this is to write down three general matrices and compute the triple product explicitly, both ways, and check that you get the same answer. A simpler, and possibly more aesthetically pleasing approach, is as follows. Note that a 2×2 matrix A determines a map f : R2 −! R2; where R2 = R×R is the ordinary cartesian product. Given v = (a; b) 2 R2, formally treat this as a column vector and define f(a; b) = Av; 1 where Av is computed in the usual way. Note that, in these terms, the product of two matrices corresponds to composition of functions, and that f determines A; indeed the top row of the matrix can be recovered from Av, where v = (1; 0) and the bottom row from Av, where v = (0; 1).
    [Show full text]
  • Constructing Zero Divisor Graphs
    Introduction/Background Our Methods Results and Conclusions References Constructing Zero Divisor Graphs Alaina Wickboldt, Louisiana State University Alonza Terry, Xavier University of Louisiana Carlos Lopez, Mississippi State University SMILE 2011 A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs Introduction/Background Our Methods Results and Conclusions References Outline Introduction/Background Ring Theory Modular Arithmetic Zero Divisor Graphs Our Methods Drawing by hand Implementing Computer Coding Results and Conclusions Results Conclusion References A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs Introduction/Background Our Methods Results and Conclusions References Joint work with: • Dr. Sandra Spiroff, University of Mississippi • Dr. Dave Chapman, Louisiana State University A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs • Construct the [AL] graph Γ(R) for R = (Zn) up to order n, for n ≤ 100. • Observe behavior and patterns of each graph. Introduction/Background Our Methods Results and Conclusions References THE PROJECT To study the various definitions of zero divisor graphs associated to rings, namely those by I. Beck [B] and D. Anderson & P. Livingston [AL]. A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs Introduction/Background Our Methods Results and Conclusions References THE PROJECT To study the various definitions of zero divisor graphs associated to rings, namely those by I. Beck [B] and D. Anderson & P. Livingston [AL]. • Construct the [AL] graph Γ(R) for R = (Zn) up to order n, for n ≤ 100. • Observe behavior and patterns of each graph. A. Wickboldt, A. Terry, C. Lopez Constructing Zero Divisor Graphs Introduction/Background Ring Theory Our Methods Modular Arithmetic Results and Conclusions Zero Divisor Graphs References Relations Between Rings and Graphs Ring Graph 2 1 < ¢¢ << ¢¢ ¢¢ << ¢¢ ¢¢ << ¢¢ ¢¢ < ¢¢ 3 0 G( ) < < Z6 << ¢¢ << << ¢¢ << Z/6Z << ¢¢ << < ¢¢ < 4 5 A.
    [Show full text]
  • 28. Com. Rings: Units and Zero-Divisors
    M7210 Lecture 28 Friday October 26, 2012 Commutative Rings III We assume all rings are commutative and have multiplicative identity (different from 0). Units and zero-divisors Definition. Let A be a commutative ring and let a ∈ A. i) a is called a unit if a 6=0A and there is b ∈ A such that ab =1A. ii) a is called a zero-divisor if a 6= 0A and there is non-zero b ∈ A such that ab = 0A. A (commutative) ring with no zero-divisors is called an integral domain. Remark. In the zero ring Z = {0}, 0Z = 1Z. This is an annoying detail that our definition must take into account, and this is why the stipulation “a 6= 0A” appears in i). The zero ring plays a very minor role in commutative algebra, but it is important nonetheless because it is a valid model of the equational theory of rings-with-identity and it is the final object in the category of rings. We exclude this ring in the discussion below (and occasionally include a reminder that we are doing so). Fact. In any (non-zero commutative) ring R the set of zero-divisors and the set of units are disjoint. Proof . Let u ∈ R \{0}. If there is z ∈ R such that uz = 0, then for any t ∈ R, (tu)z = 0. Thus, there is no t ∈ R such that tu = 1. Fact. Let R be a ring. The set U(R) of units of R forms a group. Examples. The units in Z are 1 and −1.
    [Show full text]