Emmy Noether’s theory of ideals in rings furthered Richard Dedekind’s studies of polynomic root solutions and laid the foundations for much of abstract algebra.
It is almost impossible to give a nonesoteric account of certain highly abstract and technical developments in areas of higher mathematics, such as the ring and ideal theory in modern algebra. This is so because many key developments are highly formalized and have few or no equivalents in ordinary mathematics. Thus this treatment focuses instead primarily on the status and impact of these developments.
A set of elements for which the binary operations of addition and multiplication are defined, so as to satisfy the basic axioms of commutativity, associativity, and distributivity, is known as a ring. The common integer counting numbers for a ring, as for the polynomials of all degrees N, are defined as f = f(x) = a0
+ a1x + a2x2
+ . . . + anxn
. The term “ring” derives from the operation of wrapping a segment of the lineal real number line around a circle to obtain a modular mapping. The German mathematician Ernst Eduard Kummer
+ yn
= zn
for all n. Kummer made the unwarranted assumption that a hypothesis called the unique factorization theorem is always true in rings of algebraic integers. A number in the complex field is called an algebraic integer if it satisfies an algebraic equation with rational integer coefficients not greater than 1. The algebraic integers in an algebraic number field are said to form an integral domain.
Emmy Noether.
Although Kummer’s unsuccessful effort did not arrive at a proof of Fermat’s theorem, the theory of ideal numbers that arose subsequently (through Richard Dedekind’s efforts) did make it possible to establish the general conditions under which the Fermat equation would be unsolvable by means of integers. Kummer defined what Dedekind later independently termed “ideal” numbers, such that composite numbers made from the new species satisfied Carl Friedrich Gauss’s
Gauss first applied number theory factorization concepts to the ring of all complex numbers, defined by a + bi. The further development of ideals by Dedekind arose as a result of his subsequent efforts to restore unique factorization in some algebraic domains. Faced with this problem, Dedekind had the insight that, instead of considering the single number 3, for example, one could consider the set A = 3x and B = 2 + √–5y, where both x and y are in a common set Z. The sets A and B are examples of what Dedekind calls ideals.
An integral domain is a ring in which the nonzero elements constitute a semigroup of the multiplicative semigroup. Any set equipped with an operation (such as multiplication) satisfying the conditions eg = ge = g and g–1
g = e is called a group. The integral domain Z defined by Z = x + y √–5 does not have the property of factorization into unique prime numbers. Nevertheless, as Dedekind demonstrated, it is possible to develop unique factorizations in some kind of prime quantities for general algebraic fields. If D is an integral domain, a nonempty set of D, K is called an ideal if it satisfies the following conditions: that b1
+ b2
is a part of K, that b1
– b2
is a part of K, and that r
b is a part of K, or the equivalent condition that K is a group with respect to addition.
An ideal is not itself a number; rather, it is a set of numbers qualifying as a special kind of subring of a ring. If R is a ring, a subring of R is a subset, which is a ring with respect to addition and multiplication in R. Multiplication and other operations with ideals can also be defined. Some mathematicians have ascribed the rather odd choice of the name “ideal” to a prophetic intuition of the future formal axiomatics of David Hilbert.
With Emmy Noether, the ideals first invented for use in number theory became a more fundamental tool of wider use in higher algebra and elsewhere. Noether was the daughter of Max Noether, a noted mathematician who played an important role in contributing to the Erlangen school’s development of the theory of algebraic functions. During the 1903-1904 semester, despite official opposition because of her gender, Emmy Noether was permitted to enroll as an auditor in courses at the University of Göttingen taught by Hilbert, Felix Klein, and Hermann Minkowski. Hilbert and Klein recognized her potential, and, after further studies at the University of Erlangen under Paul Gordan, she received her Ph.D. in 1907. Her thesis extended Gordan’s finiteness problem of invariants for two variables. After 1911, Noether turned toward Hilbert’s formal and abstract methodology. In 1915, Hilbert and Klein invited her to work at Göttingen. She stayed there as an untenured associate professor until she was dismissed by the Nazi regime in 1933.
In addition to Dedekind’s work, the research of Ernst Steinitz
Noether developed a more general theory of ideals using the formal axiomatic basis of Hilbert to include all prior examples as well as extend the range of ideals. The chief innovation of this work is the so-called ascending-chain axiom, which states that a chain of ideals, a1
, a2
, a3
, . . . an
, necessarily comes to an end after a finite number of steps if each term a1
includes the preceding term a1
–1 as a so-called proper part. Noether’s ascending-chain condition can be shown to be equivalent to Hilbert’s earlier theorem of the finite ideal basis.
More generally, through this and several later papers, Noether showed how it is possible to derive the polynomial ideals of Kummer in the same axiomatic fashion and to retrieve all of Dedekind’s classical results on ideals in algebraic number fields, the latter requiring the condition of the ring to be “integrally closed.” A closed ring is a set of numbers where there is no means to obtain a number outside the set by any means of addition, multiplication, or the like.
Many of Noether’s later discoveries were notably furthered by her students of the “Noether school,” including Wolfgang Krull, Gottfried Koethe, and notably Bartel Leendert van der Waerden, whose two-volume text on advanced algebra remained the standard for many decades. The widespread modern tendency of viewing algebraic structures as groups with operators can be traced directly to Noether’s publications and those of her associates. Instead of operating with definite or even formal expressions, the simple properties of the operations, for example, of addition and multiplication to which they lend themselves, were reformulated as initial axioms, forming the basis for further deductions. Noether’s particular strength, acknowledged by many of her contemporaries and students, lay in her uncanny ability to operate abstractly with concepts alone within the framework of her drive toward axiomatic purity.
The notion of ideals in other new versions played a decisive role in Noether’s theory of noncommutative algebras and their representation. Although most common operations with numbers in multiplication are commutative (a
b = b
a), in the more general interpretation of multiplication provided by abstract algebra, noncommutative algebra was derived abstractly by Noether between 1924 and 1928. In Noether’s abstract approach, traditional calculational tools, such as matrices and determinants, used so successfully by her contemporary I. J. Schur in 1926, are discarded entirely in favor of purely abstract concepts. In intensive cooperation with colleagues, Noether carefully investigated the detailed structure of noncommutative algebras and applied this theory further to the ordinary commutative number fields and their arithmetics by means of her so-called cross-product method. The important influence of many of these later efforts by Noether is clearly visible in the influential textbook by physicist-intuitionist mathematician Hermann Weyl Gruppentheorie und Quantenmechanik (1928; The Theory of Groups and Quantum Mechanics, 1950), who also made notable contributions to the algebraic theory of numbers.
The dissemination and development of Noether’s work was ensured by the twelve doctoral students under her directorship as well as by a host of French, German, and Soviet mathematicians with whom she frequently visited and corresponded. By eliminating some of Noether’s original axioms, van der Waerden and Emil Artin later obtained further generalizations of her ideal theory. The most far-reaching generalizations of the Dedekind-Noether theory of ideals were produced by German mathematician, logician, and philosopher Paul Lorenzen. A special subtype of ideals has been named Noetherian in her honor.
As the Nicolas Bourbaki group notes in Éléments de mathématique (1939; elements of mathematics), the wider penetration of general algebraic ideas and methods into the diverse network of mathematical and physical subdisciplines became possible only after the publications of Noether. One can judge the effects of Noether’s work on the theory of ring-ideals by comparing the organization, methods, and focus of van der Waerden’s theory to modern algebra of 1930 with the pre-Noetherian treatment of algebra.
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Dick, Auguste. Emmy Noether, 1882-1935. Translated by Heide I. Blocher. 1970. Reprint. Cambridge, Mass.: Birkhäuser Boston, 1981. The earliest biography of Noether in a single volume. Authoritative. -
Jones, Burton W. An Introduction to Modern Algebra. New York: Macmillan, 1975. Gives a general account of modern ring and ideal theory. -
Robinson, Abraham. Numbers and Ideals: An Introduction to Some Basic Concepts of Algebra and Number Theory. San Francisco: Holden Day, 1965. A unique introductory text treating ideals with minimum prerequisites. -
Srinivasan, Bhama, and Judith Sally. Emmy Noether in Bryn Mawr: Proceedings of a Symposium. New York: Springer-Verlag, 1983. Presentation of eulogies and studies gives important perspective on Noether’s accomplishments. -
Weil, André. Number Theory: An Approach Through History from Hammurapi to Legendre. 1984. Reprint. Cambridge, Mass.: Birkhäuser Boston, 2000. A unique presentation of abstract arithmetic by one of the founders of the Bourbaki circle.
Brouwer Develops Intuitionist Foundations of Mathematics
Zermelo Undertakes Comprehensive Axiomatization of Set Theory
Steinitz Inaugurates Modern Abstract Algebra
Noether Shows the Equivalence of Symmetry and Conservation