Algebra Commons^™

Splitting Fields And Periods Of Fibonacci Sequences Modulo Primes, Sanjai Gupta, Parousia Rockstroh '08, Francis E. Su

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We consider the period of a Fibonacci sequence modulo a prime and provide an accessible, motivated treatment of this classical topic using only ideas from linear and abstract algebra. Our methods extend to general recurrences with prime moduli and provide some new insights. And our treatment highlights a nice application of the use of splitting fields that might be suitable to present in an undergraduate course in abstract algebra or Galois theory.

Squaring, Cubing, And Cube Rooting, Arthur T. Benjamin

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I still recall my thrill and disappointment when I read Mathematical Carnival, by Martin Gardner. I was thrilled because, as my high school teacher had recommended, mathematics was presented in a playful way that I had never seen before. I was disappointed because it contained a formula that I thought I had "invented" a few years earlier. I have always had a passion for mental calculation, and the following formula appears in Gardner's chapter on "Lightning Calculators." It was used by the mathematician A. C. Aitken to mentally square large numbers.

Recognizing Graph Theoretic Properties With Polynomial Ideals, Jesus A. De Loera, Christopher J. Hillar, Peter N. Malkin, Mohamed Omar

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Many hard combinatorial problems can be modeled by a system of polynomial equations. N. Alon coined the term polynomial method to describe the use of nonlinear polynomials when solving combinatorial problems. We continue the exploration of the polynomial method and show how the algorithmic theory of polynomial ideals can be used to detect k-colorability, unique Hamiltonicity, and automorphism rigidity of graphs. Our techniques are diverse and involve Nullstellensatz certificates, linear algebra over finite fields, Gröbner bases, toric algebra, convex programming, and real algebraic geometry.

A Look At Biseparating Maps From An Algebraic Point Of View, Melvin Henriksen, Frank A. Smith

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In [ABN], Araujo, Beckenstein, and Narici add the capstone to a series of papers by several groups of authors by showing that if ρ is a biseparating map between two algebras of all real or complex-valued functions on realcompact spaces, then it is a continuous multiple of an isomorphism between these rings. Their proof uses relatively powerful analytic and topological techniques. In what follows, the extent to which such a result can be generalized to a wider class of algebras using algebraic techniques is investigated. We are unable, however to obtain the main result of [ABN] using these techniques.

Sensible Rules For Remembering Duals -- The S-O-B Method, Arthur T. Benjamin

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We present a natural motivation and simple mnemonic for creating the dual LP of any linear programing problem.

Lattice-Ordered Algebras That Are Subdirect Products Of Valuation Domains, Melvin Henriksen, Suzanne Larson, Jorge Martinez, R. G. Woods

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An f-ring (i.e., a lattice-ordered ring that is a subdirect product of totally ordered rings) A is called an SV-ring if A/P is a valuation domain for every prime ideal P of A. If M is a maximal ℓ-ideal of A , then the rank of A at M is the number of minimal prime ideals of A contained in M, rank of A is the sup of the ranks of A at each of its maximal ℓ-ideals. If the latter is a positive integer, then A is said to have finite rank, and if A …

The Space Of Minimal Prime Ideals Of C(X) Need Not Be Basically Disconnected, Alan Dow, Melvin Henriksen, Ralph Kopperman, J. Vermeer

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Problems posed twenty and twenty-five years ago by M. Henriksen and M. Jerison are solved by showing that the space of minimal prime ideals of the ring C(X) of continuous real-valued functions on a compact (Hausdorff) space need not be basically disconnected-or even an F-space.

Locally Finite Families, Completely Separated Sets And Remote Points, Melvin Henriksen, Thomas J. Peters

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It is shown that if X is a nonpseudocompact space with a σ-locally finite π-base, then X has remote points. Within the class of spaces possessing a σ-locally finite π-base, this result extends the work of Chae and Smith, because their work utilized normality to achieve complete separation. It provides spaces which have remote points, where the spaces do not satisfy the conditions required in the previous works by Dow, by van Douwen, by van Mill, or by Peters.

The lemma: "Let X be a space and let {Cε: € < α} be a locally finite family of cozero sets of X. Let {Zε: € < α } be a family of zero sets of X such that for each € < α, Zε с Cε. Then ∪_ε<α Zε is completely separated from X/∪εCε", is a fundamental …

Some Properties Of Positive Derivations On F-Rings, Melvin Henriksen, Frank A. Smith

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Throughout A denotes an f-ring; that is, a lattice-ordered ring that is a subdirect union of totally ordered rings. We let D(A) denote the set of derivations D: A --> A such that a ≥ 0 implies Da ≥ 0, and we call such derivations positive. In [CDK], P. Coleville, G. Davis, and K. Keimel initiated a study of positive derivations on f-rings. Their main results are (i) D ε D(A) and A archimedean imply D = 0, and (ii) if A has an identity element 1 and a is the supremum of a set …

A Summary Of Results On Order-Cauchy Completions Of Rings And Vector Lattices Of Continuous Functions, Melvin Henriksen

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This paper is a summary of joint research by F. Dashiell, A. Hager and the present author. Proofs are largely omitted. A complete version will appear in the Canadian Journal of Mathematics. It is devoted to a study of sequential order-Cauchy convergence and the associated completion in vector lattices of continuous functions. Such a completion for lattices C(X) is related to certain topological properties of the space X and to ring properties of C(X). The appropriate topological condition on the space X equivalent to this type of completeness for the lattice C(X) was first identified for compact spaces X in …

An Algebraic Characterization Of The Freudenthal Compactification For A Class Of Rimcompact Spaces, Melvin Henriksen

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Throughout C(X) will denote the ring of all continuous real-valued functions on a Tychonoff space X, and C*(X) will denote the subring of bounded elements of C(X). The real line is denoted by R, and N denotes the (discrete) subspace of positive integers. A subset S of X such that the map f → f|_s is an epimorphism of C(X) (resp. C*(X)) is said to be C-embedded (resp. C*-embedded) in X. As is well-known, every f Є C*(X) has a unique continuous extension βf over its Stone-Čech compactification βX [GJ, Chapter 6]. That is, X is …

A Simple Characterization Of Commutative Rings Without Maximal Ideals, Melvin Henriksen

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In a course in abstract algebra in which the instructor presents a proof that each ideal in a ring with identity is contained in a maximal ideal, it is customary to give an example of a ring without maximal ideals.

Some Remarks About Elementary Divisor Rings, Leonard Gillman, Melvin Henriksen

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By a slight modification of Kaplansky's argument, we find that the condition on zero-divisors can be replaced by the hypothesis that S be an Hermite ring (i.e., every matrix over S can be reduced to triangular form). This is an improvement, since, in any case, it is necessary that S be an Hermite ring, while, on the other hand, it is not necessary that all zero-divisors be in the radical. In fact, we show that every regular commutative ring with identity is adequate. However, the condition that S be adequate is not necessary either.

We succeed in obtaining a necessary …

Rings Of Continuous Functions In Which Every Finitely Generated Ideal Is Principal, Leonard Gillman, Melvin Henriksen

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The outline of our present paper is as follows. In §1, we collect some preliminary definitions and results. §2 inaugurates the study of F-rings and F-spaces (i.e., those spaces X for which C(X) is an F-ring).

The space of reals is not an F-space; in fact, a metric space is an F-space if and only if it is discrete. On the other hand, if X is any locally compact, σ-compact space (e.g., the reals), then βX-X is an F-space. Examples of necessary and sufficient conditions for an arbitrary completely regular space to be an F-space are:

(i) for every f …

Some Remarks On Elementary Divisor Rings Ii, Melvin Henriksen

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A commutative ring S with identity element 1 is called an elementary divisor ring (resp. Hermite ring) if for every matrix A over S there exist nonsingular matrices P, Q such that PAQ (resp. AQ) is a diagonal matrix (resp. triangular matrix). It is clear that every elementary divisor ring is an Hermite ring, and that every Hermite ring is an F-ring (that is, a commutative ring with identity in which all finitely generated ideals are principal).

Concerning Rings Of Continuous Functions, Leonard Gillman, Melvin Henriksen

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The present paper deals with two distinct, though related, questions, concerning the ring C(X, R) of all continuous real-valued functions on a completely regular topological space X.

The first of these, treated in §§1-7, is the study of what we call P-spaces -- those spaces X such that every prime ideal of the ring C(X, R) is a maximal ideal. The background and motivation for this problem are set forth in §1. The results consist of a number of theorems concerning prime ideals of the ring C(X, R) in general, as well as a series of characterizations of P-spaces in …

On A Theorem Of Gelfand And Kolmogoroff Concerning Maximal Ideals In Rings Of Continuous Functions, Leonard Gillman, Melvin Henriksen, Meyer Jerison

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This paper deals with a theorem of Gelfand and Kolmogoroff concerning the ring C= C(X, R) of all continuous real-valued functions on a completely regular topological space X, and the subring C* = C*(X, R) consisting of all bounded functions in C. The theorem in question yields a one-one correspondence between the maximal ideals of C and those of C*; it is stated without proof in [2]. Here we supply a proof (§2), and we apply the theorem to three problems previously considered by Hewitt in [5].

Our first result (§3) consists of two simple constructions of the Q-space vX. …

On The Continuity Of The Real Roots Of An Algebraic Equation, Melvin Henriksen, John R. Isbell

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It is well known that the root of an algebraic equation is a continuous multiple-valued function of its coefficients [5, p. 3]. However, it is not necessarily true that a root can be given by a continuous single-valued function. A complete solution of this problem has long been known in the case where the coefficients are themselves polynomials in a complex variable [3, chap. V]. For most purposes the concept of the Riemann surface enables one to bypass the problem. However, in the study of the ideal structure of rings of continuous functions, the general problem must be met directly. …

On The Prime Ideals Of The Ring Of Entire Functions, Melvin Henriksen

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Let R be the ring of entire functions, and let K be the complex field. In an earlier paper [6], the author investigated the ideal structure of R, particular attention being paid to the maximal ideals. In 1946, Schilling [9, Lemma 5] stated that every prime ideal of R is maximal. Recently, I. Kaplansky pointed out to the author (in conversation) that this statement is false, and constructed a non maximal prime ideal of R (see Theorem 1(a), below). The purpose of the present paper is to investigate these nonmaximal prime ideals and their residue class fields. The author is …

On The Ideal Structure Of The Ring Of Entire Functions, Melvin Henriksen

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Let R be the ring of entire functions, and let K be the complex field. The ring R consists of all functions from K to K differentiable everywhere (in the usual sense).

The algebraic structure of the ring of entire functions seems to have been investigated extensively first by O. Helmer [1].

The ideals of R are herein classified as in [2]: an ideal I is called fixed if every function in it vanishes at at least one common point; otherwise, I is called free. The structure of the fixed ideals was determined in [1]. The structure of the …