Algebra Commons^™

K-Theory And Steenrod Homology: Applications To The Brown-Douglas-Fillmore Theory Of Operator Algebras, Jerome Kaminker, Claude Schochet

Mathematics Faculty Research Publications

The remarkable work of L. G. Brown, R. Douglas and P. Fillmore on operators with compact self-commutators once again ties together algebraic topology and operator theory. This paper gives a comprehensive treatment of certain aspects of that connection and some adjacent topics. In anticipation that both operator theorists and topologists may be interested in this work, additional background material is included to facilitate access.

K₁ Of The Compact Operators Is Zero, L. G. Brown, Claude Schochet

Mathematics Faculty Research Publications

We prove that K₁ of the compact operators is zero. This theorem has the following operator-theoretic formulation: any invertible operator of the form (identity) + (compact) is the product of (at most eight) multiplicative commutators (A_jB_jA_j⁻¹B_j⁻¹)^±1, where each B_j is of the form (identity) + (compact). The proof uses results of L. G. Brown, R. G. Douglas, and P. A. Fillmore on essentially normal operators and a theorem of A. Brown and C. Pearcy on multiplicative …

Approximation Of Compact Homogeneous Maps, John R. Hubbard

Department of Math & Statistics Faculty Publications

Within the clasp of continuous homogeneous maps between Banach spaces, it is proved that every compact map can be uniformly approximated by finite-rank maps. This result is obtained by means of the classical metric projection on Banach spaces.

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

All HMC Faculty Publications and Research

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.

Steenrod Homology And Operator Algebras, Jerome Kaminker, Claude Schochet

Mathematics Faculty Research Publications

The recent work of Larry Brown, R. G. Douglas, and Peter Fillmore on operator algebras has created a new bridge between functional analysis and algebraic topology. This note constitutes an effort to make that bridge more concrete.

The Solution Of Equations By Radicals, Stephen Davis

Student Scholarship

This paper investigates the algebraic conjecture that the general equation of degree n is not solvable by radicals if n is greater than four. While the author does not intend to provide a new rigorous proof, the work serves to organize and present the fundamental mathematical concepts necessary to understand the proof for an undergraduate audience. The historical context of the problem is explored, noting the early achievements in solving cubic and quartic equations during the sixteenth century and the later contributions of mathematicians like Lagrange, who attempted to find a general reduction method for the quintic. The ultimate solution …

A Comparative Investigation Of The Effects Of Frequent Testing Upon Achievement In Secondary Advanced Algebra, John Thomas Fullerton

All Master's Theses

Relatively speaking, few studies have concerned themselves with the problem of frequent testing, and as Keys pointed out, empirical evidence, uncomplicated by differences in the amount of testing material employed, on the effects of frequent testing is, at best, scarce (14:427). Also many studies used tests and test results for direct instruction, thus introducing additional variables. Furthermore, the choice of subjects and disciplines has been limited, the better part being taken from college psychology and sociology classes or high school science classes. This investigation was not an attempt to modify previous experiments, nor was it an attempt to identify which …

Simple Algebraic Extensions And Characteristics Polynomials, Don Stokes

Journal of the Arkansas Academy of Science

No abstract provided.

On Semigroup Rings, Lawrence Paul Runyan

All Master's Theses

Some of the properties of semigroup rings are described, illustrated and proved in this study. In particular, the central problem focuses on how various properties of the semigroup and ring are reflected in the resulting semigroup ring.

An Investigation Of The Range Of A Boolean Function, Norman H. Eggert, Jr.

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

The purpose of this section is to define a boolean algebra and to determine some of the important properties of it.

A boolean algebra is a set B with two binary operations, join and meet, denoted by + and juxtaposition respectively, and a unary operation, complementation, denoted by ', which satisfy the following axioms:

(1) for all a,b ∑ B (that is, for all a,b elements of B) a + b = b + a and a b = b a, (the commutative laws),

(2) for all a,b,c ∑ B, a + b c =(a + b) (a + b) …

The Uniform Right Ideals Of A Ring Of Matrices, Gerald Joseph Janusz

Bachelors’ Theses

The concept of uniform right ideals plays an important part in the structure tbeory of prime rings and semi-prime rings as developed by Goldie in [1] and (=[2]. Very few examples of uniform right ideals appear in the literature, however. In section one, we obtain the form of these right ideals in a matrix ring, I_n, where I is a right Ore domain with identity. The existence of the identity involves some loss of generality, but greatly simplifies the approach used in this paper.

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

All HMC Faculty Publications and Research

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

All HMC Faculty Publications and Research

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

All HMC Faculty Publications and Research

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

All HMC Faculty Publications and Research

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

All HMC Faculty Publications and Research

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

All HMC Faculty Publications and Research

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

All HMC Faculty Publications and Research

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

All HMC Faculty Publications and Research

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 …

Sylvester’S Method Of Elimination, John D. Gardner

Bachelors’ Theses

To understand and appreciate Sylvester's method of elimination, it is necessary to review the most important points in the general field of algebraic elimination. As the information from which this thesis was written was derived from no particular text, specific references will, for the most part, be omitted. The titles of the books referred to are to be found in a bibliography placed at the back of the thesis.

Arithmetical Progressions, Frances Agnes Scherkenbach

Bachelors’ Theses

Of the origin and the methods of treatment very little is known. The arithmetic and geometric progression first attracted attention after the Greeks had brought into prominence the harmonic series. The term ’series’ is derived from the Greek, while the term’ progression’ is derived from the Latin. The latter term was used most prominently until the seventeenth century, during which period the writers seemed to prefer the term "series" which is now mosy [sic] generally used.

The Theory Of Exponents, Olive Marie Lass

Bachelors’ Theses

When one speaks of Algebra, with its laws and operations, one asks the question - Is it an ancient subject, and how old is it? In answer to this question I find it is necessary to go back to the years before Christ.

The Study Of The Curves Of The Equations Having The Type Form X^N + Y^N = A^N, Lena S. Reif

Bachelors’ Theses

The object of this thesis is to determine the general properties of the family of curves, x^n + y^n = a^n from some of the specific curves in that family. The first curve to be considered is the one in which a = 4 and n =1. x+y = 4

Graphic Methods For Solving Algebraic Equations, L. Tremaine Dunlap

Bachelors’ Theses

No abstract provided.

Student Algebra Notebook, Jack Thorington, Joseph S. Winter

Winter Letterbooks

Jack Thorington's Algebra notebook. Contains practice algebra problems.