6. Localization
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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. -
Prime Ideals in Polynomial Rings in Several Indeterminates
PROCEEDINGS OF THE AMIERICAN MIATHEMIIATICAL SOCIETY Volume 125, Number 1, January 1997, Pages 67 -74 S 0002-9939(97)03663-0 PRIME IDEALS IN POLYNOMIAL RINGS IN SEVERAL INDETERMINATES MIGUEL FERRERO (Communicated by Ken Goodearl) ABsTrRAcr. If P is a prime ideal of a polynomial ring K[xJ, where K is a field, then P is determined by an irreducible polynomial in K[x]. The purpose of this paper is to show that any prime ideal of a polynomial ring in n-indeterminates over a not necessarily commutative ring R is determined by its intersection with R plus n polynomials. INTRODUCTION Let K be a field and K[x] the polynomial ring over K in an indeterminate x. If P is a prime ideal of K[x], then there exists an irreducible polynomial f in K[x] such that P = K[x]f. This result is quite old and basic; however no corresponding result seems to be known for a polynomial ring in n indeterminates x1, ..., xn over K. Actually, it seems to be very difficult to find some system of generators for a prime ideal of K[xl,...,Xn]. Now, K [x1, ..., xn] is a Noetherian ring and by a converse of the principal ideal theorem for every prime ideal P of K[x1, ..., xn] there exist n polynomials fi, f..n such that P is minimal over (fi, ..., fn), the ideal generated by { fl, ..., f}n ([4], Theorem 153). Also, as a consequence of ([1], Theorem 1) it follows that any prime ideal of K[xl, X.., Xn] is determined by n polynomials. -
Group Homomorphisms
1-17-2018 Group Homomorphisms Here are the operation tables for two groups of order 4: · 1 a a2 + 0 1 2 1 1 a a2 0 0 1 2 a a a2 1 1 1 2 0 a2 a2 1 a 2 2 0 1 There is an obvious sense in which these two groups are “the same”: You can get the second table from the first by replacing 0 with 1, 1 with a, and 2 with a2. When are two groups the same? You might think of saying that two groups are the same if you can get one group’s table from the other by substitution, as above. However, there are problems with this. In the first place, it might be very difficult to check — imagine having to write down a multiplication table for a group of order 256! In the second place, it’s not clear what a “multiplication table” is if a group is infinite. One way to implement a substitution is to use a function. In a sense, a function is a thing which “substitutes” its output for its input. I’ll define what it means for two groups to be “the same” by using certain kinds of functions between groups. These functions are called group homomorphisms; a special kind of homomorphism, called an isomorphism, will be used to define “sameness” for groups. Definition. Let G and H be groups. A homomorphism from G to H is a function f : G → H such that f(x · y)= f(x) · f(y) forall x,y ∈ G. -
Modules and Vector Spaces
Modules and Vector Spaces R. C. Daileda October 16, 2017 1 Modules Definition 1. A (left) R-module is a triple (R; M; ·) consisting of a ring R, an (additive) abelian group M and a binary operation · : R × M ! M (simply written as r · m = rm) that for all r; s 2 R and m; n 2 M satisfies • r(m + n) = rm + rn ; • (r + s)m = rm + sm ; • r(sm) = (rs)m. If R has unity we also require that 1m = m for all m 2 M. If R = F , a field, we call M a vector space (over F ). N Remark 1. One can show that as a consequence of this definition, the zeros of R and M both \act like zero" relative to the binary operation between R and M, i.e. 0Rm = 0M and r0M = 0M for all r 2 R and m 2 M. H Example 1. Let R be a ring. • R is an R-module using multiplication in R as the binary operation. • Every (additive) abelian group G is a Z-module via n · g = ng for n 2 Z and g 2 G. In fact, this is the only way to make G into a Z-module. Since we must have 1 · g = g for all g 2 G, one can show that n · g = ng for all n 2 Z. Thus there is only one possible Z-module structure on any abelian group. • Rn = R ⊕ R ⊕ · · · ⊕ R is an R-module via | {z } n times r(a1; a2; : : : ; an) = (ra1; ra2; : : : ; ran): • Mn(R) is an R-module via r(aij) = (raij): • R[x] is an R-module via X i X i r aix = raix : i i • Every ideal in R is an R-module. -
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 . -
Math 296. Homework 3 (Due Jan 28) 1
Math 296. Homework 3 (due Jan 28) 1. Equivalence Classes. Let R be an equivalence relation on a set X. For each x ∈ X, consider the subset xR ⊂ X consisting of all the elements y in X such that xRy. A set of the form xR is called an equivalence class. (1) Show that xR = yR (as subsets of X) if and only if xRy. (2) Show that xR ∩ yR = ∅ or xR = yR. (3) Show that there is a subset Y (called equivalence classes representatives) of X such that X is the disjoint union of subsets of the form yR for y ∈ Y . Is the set Y uniquely determined? (4) For each of the equivalence relations from Problem Set 2, Exercise 5, Parts 3, 5, 6, 7, 8: describe the equivalence classes, find a way to enumerate them by picking a nice representative for each, and find the cardinality of the set of equivalence classes. [I will ask Ruthi to discuss this a bit in the discussion session.] 2. Pliability of Smooth Functions. This problem undertakes a very fundamental construction: to prove that ∞ −1/x2 C -functions are very soft and pliable. Let F : R → R be defined by F (x) = e for x 6= 0 and F (0) = 0. (1) Verify that F is infinitely differentiable at every point (don’t forget that you computed on a 295 problem set that the k-th derivative exists and is zero, for all k ≥ 1). −1/x2 ∞ (2) Let ϕ : R → R be defined by ϕ(x) = 0 for x ≤ 0 and ϕ(x) = e for x > 0. -
The General Linear Group
18.704 Gabe Cunningham 2/18/05 [email protected] The General Linear Group Definition: Let F be a field. Then the general linear group GLn(F ) is the group of invert- ible n × n matrices with entries in F under matrix multiplication. It is easy to see that GLn(F ) is, in fact, a group: matrix multiplication is associative; the identity element is In, the n × n matrix with 1’s along the main diagonal and 0’s everywhere else; and the matrices are invertible by choice. It’s not immediately clear whether GLn(F ) has infinitely many elements when F does. However, such is the case. Let a ∈ F , a 6= 0. −1 Then a · In is an invertible n × n matrix with inverse a · In. In fact, the set of all such × matrices forms a subgroup of GLn(F ) that is isomorphic to F = F \{0}. It is clear that if F is a finite field, then GLn(F ) has only finitely many elements. An interesting question to ask is how many elements it has. Before addressing that question fully, let’s look at some examples. ∼ × Example 1: Let n = 1. Then GLn(Fq) = Fq , which has q − 1 elements. a b Example 2: Let n = 2; let M = ( c d ). Then for M to be invertible, it is necessary and sufficient that ad 6= bc. If a, b, c, and d are all nonzero, then we can fix a, b, and c arbitrarily, and d can be anything but a−1bc. This gives us (q − 1)3(q − 2) matrices. -
The Structure Theory of Complete Local Rings
The structure theory of complete local rings Introduction In the study of commutative Noetherian rings, localization at a prime followed by com- pletion at the resulting maximal ideal is a way of life. Many problems, even some that seem \global," can be attacked by first reducing to the local case and then to the complete case. Complete local rings turn out to have extremely good behavior in many respects. A key ingredient in this type of reduction is that when R is local, Rb is local and faithfully flat over R. We shall study the structure of complete local rings. A complete local ring that contains a field always contains a field that maps onto its residue class field: thus, if (R; m; K) contains a field, it contains a field K0 such that the composite map K0 ⊆ R R=m = K is an isomorphism. Then R = K0 ⊕K0 m, and we may identify K with K0. Such a field K0 is called a coefficient field for R. The choice of a coefficient field K0 is not unique in general, although in positive prime characteristic p it is unique if K is perfect, which is a bit surprising. The existence of a coefficient field is a rather hard theorem. Once it is known, one can show that every complete local ring that contains a field is a homomorphic image of a formal power series ring over a field. It is also a module-finite extension of a formal power series ring over a field. This situation is analogous to what is true for finitely generated algebras over a field, where one can make the same statements using polynomial rings instead of formal power series rings. -
Formal Power Series - Wikipedia, the Free Encyclopedia
Formal power series - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Formal_power_series Formal power series From Wikipedia, the free encyclopedia In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates. This perspective contrasts with that of power series, whose variables designate numerical values, and which series therefore only have a definite value if convergence can be established. Formal power series are often used merely to represent the whole collection of their coefficients. In combinatorics, they provide representations of numerical sequences and of multisets, and for instance allow giving concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved; this is known as the method of generating functions. Contents 1 Introduction 2 The ring of formal power series 2.1 Definition of the formal power series ring 2.1.1 Ring structure 2.1.2 Topological structure 2.1.3 Alternative topologies 2.2 Universal property 3 Operations on formal power series 3.1 Multiplying series 3.2 Power series raised to powers 3.3 Inverting series 3.4 Dividing series 3.5 Extracting coefficients 3.6 Composition of series 3.6.1 Example 3.7 Composition inverse 3.8 Formal differentiation of series 4 Properties 4.1 Algebraic properties of the formal power series ring 4.2 Topological properties of the formal power series -
The Fundamental Homomorphism Theorem
Lecture 4.3: The fundamental homomorphism theorem Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4120, Modern Algebra M. Macauley (Clemson) Lecture 4.3: The fundamental homomorphism theorem Math 4120, Modern Algebra 1 / 10 Motivating example (from the previous lecture) Define the homomorphism φ : Q4 ! V4 via φ(i) = v and φ(j) = h. Since Q4 = hi; ji: φ(1) = e ; φ(−1) = φ(i 2) = φ(i)2 = v 2 = e ; φ(k) = φ(ij) = φ(i)φ(j) = vh = r ; φ(−k) = φ(ji) = φ(j)φ(i) = hv = r ; φ(−i) = φ(−1)φ(i) = ev = v ; φ(−j) = φ(−1)φ(j) = eh = h : Let's quotient out by Ker φ = {−1; 1g: 1 i K 1 i iK K iK −1 −i −1 −i Q4 Q4 Q4=K −j −k −j −k jK kK j k jK j k kK Q4 organized by the left cosets of K collapse cosets subgroup K = h−1i are near each other into single nodes Key observation Q4= Ker(φ) =∼ Im(φ). M. Macauley (Clemson) Lecture 4.3: The fundamental homomorphism theorem Math 4120, Modern Algebra 2 / 10 The Fundamental Homomorphism Theorem The following result is one of the central results in group theory. Fundamental homomorphism theorem (FHT) If φ: G ! H is a homomorphism, then Im(φ) =∼ G= Ker(φ). The FHT says that every homomorphism can be decomposed into two steps: (i) quotient out by the kernel, and then (ii) relabel the nodes via φ. G φ Im φ (Ker φ C G) any homomorphism q i quotient remaining isomorphism process (\relabeling") G Ker φ group of cosets M. -
Homomorphisms and Isomorphisms
Lecture 4.1: Homomorphisms and isomorphisms Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4120, Modern Algebra M. Macauley (Clemson) Lecture 4.1: Homomorphisms and isomorphisms Math 4120, Modern Algebra 1 / 13 Motivation Throughout the course, we've said things like: \This group has the same structure as that group." \This group is isomorphic to that group." However, we've never really spelled out the details about what this means. We will study a special type of function between groups, called a homomorphism. An isomorphism is a special type of homomorphism. The Greek roots \homo" and \morph" together mean \same shape." There are two situations where homomorphisms arise: when one group is a subgroup of another; when one group is a quotient of another. The corresponding homomorphisms are called embeddings and quotient maps. Also in this chapter, we will completely classify all finite abelian groups, and get a taste of a few more advanced topics, such as the the four \isomorphism theorems," commutators subgroups, and automorphisms. M. Macauley (Clemson) Lecture 4.1: Homomorphisms and isomorphisms Math 4120, Modern Algebra 2 / 13 A motivating example Consider the statement: Z3 < D3. Here is a visual: 0 e 0 7! e f 1 7! r 2 2 1 2 7! r r2f rf r2 r The group D3 contains a size-3 cyclic subgroup hri, which is identical to Z3 in structure only. None of the elements of Z3 (namely 0, 1, 2) are actually in D3. When we say Z3 < D3, we really mean is that the structure of Z3 shows up in D3. -
Algebra 557: Weeks 3 and 4
Algebra 557: Weeks 3 and 4 1 Expansion and Contraction of Ideals, Primary ideals. Suppose f : A B is a ring homomorphism and I A,J B are ideals. Then A can be thought→ of as a subring of B. We denote by⊂ Ie ( expansion⊂ of I) the ideal IB = f( I) B of B, and by J c ( contraction of J) the ideal J A = f − 1 ( J) A. The following are easy to verify: ∩ ⊂ I Iec, ⊂ J ce J , ⊂ Iece = Ie , J cec = J c. Since a subring of an integral domain is also an integral domain, and for any prime ideal p B, A/pc can be seen as a subring of B/p we have that ⊂ Theorem 1. The contraction of a prime ideal is a prime ideal. Remark 2. The expansion of a prime ideal need not be prime. For example, con- sider the extension Z Z[ √ 1 ] . Then, the expansion of the prime ideal (5) is − not prime in Z[ √ 1 ] , since 5 factors as (2 + i)(2 i) in Z[ √ 1 ] . − − − Definition 3. An ideal P A is called primary if its satisfies the property that for all x, y A, xy P , x P⊂, implies that yn P, for some n 0. ∈ ∈ ∈ ∈ ≥ Remark 4. An ideal P A is primary if and only if all zero divisors of the ring A/P are nilpotent. Since⊂ thsi propert is stable under passing to sub-rings we have as before that the contraction of a primary ideal remains primary. Moreover, it is immediate that Theorem 5.