## Idealtheorie in Ringbereichen.

Berlin: Springer, 1921.

First edition, very rare offprint, of one of the most important works of “the most significant creative mathematical genius thus far produced since the higher education of women began” (Einstein, obituary in The New York Times, May 5, 1935). In the same year, but before she died, Norbert Wiener wrote: “Miss Noether is … the greatest woman mathematician who has ever lived; and the greatest woman scientist of any sort now living, and a scholar at least on the plane of Madame Curie.” “The prominent algebraist Irving Kaplansky called Emmy Noether the ‘mother of modern algebra.’ The equally prominent Saunders MacLane asserted that ‘abstract algebra,’ as a conscious discipline, starts with Noether’s 1921 paper ‘Ideal Theory in Rings’ [the offered paper]. Hermann Weyl claimed that she ‘changed the face of algebra by her work’” (Kleiner, p. 91). During the period from 1920 to 1926, she attracted numerous mathematicians and students – she was the doctoral advisor for ten – to her research program and she became a leader in the development of modern abstract algebra” (Grolier). A ring is an algebraic object which shares some, but not all, of the properties of the integers (whole numbers): it has addition and multiplication (and the result of these operations does not depend on the order in which they are performed), there are 0 and 1, but division is not usually possible and indeed the product of non-zero elements can be zero. The integers form the simplest example of a ring, but many other examples arise from geometry and number theory, as well as other areas of mathematics. In the present paper Noether extends to the general setting of a ring some well-known properties of the factorization of integers into products of prime numbers. It turns out that this cannot usually be done with the elements of the ring – rather it is the ‘ideals’ of the ring which enjoy good factorization properties (an ideal is a subset of the ring with certain properties – see below). “Formulating geometry and number theory in the language of rings is currently a massive mathematical operation, and Noether’s work is a turning point in that endeavour” (Gray, p. 295). No copies located on OCLC or in auction records.

According to Emmy Noether’s student and successor Bartel van der Waerden, “the essence of Noether’s mathematical credo is contained in the following maxim: ‘All relations between numbers, functions and operations become perspicuous, capable of generalization, and truly fruitful after being detached from specific examples, and traced back to conceptual connections.’ We identify these ideas with the abstract, axiomatic approach in mathematics. They sound commonplace to us. But they were not so in Noether’s time. In fact, they are commonplace today in considerable part because of her work.

“Algebra in the nineteenth century was concrete by our standards. It was connected in one way or another with real or complex numbers. For example, some of the great contributors to algebra in the nineteenth century, mathematicians whose works shaped the algebra of the twentieth century, were Gauss, Galois, Jordan, Kronecker, Dedekind, and Hilbert. Their algebraic works dealt with quadratic forms, cyclotomy, field extensions, permutation groups, ideals in rings of integers of algebraic number fields, and invariant theory. All of these works were related in one way or another to real or complex numbers.

“Moreover, even these important works in algebra were viewed in the nineteenth century, in the overall mathematical scheme, as secondary. The primary mathematical fields in that century were analysis (complex analysis, differential equations, real analysis), and geometry (projective, non-euclidean, differential, and algebraic). But after the work of Noether and others in the 1920s, algebra became central in mathematics …

“Noether contributed to the following major areas of algebra: invariant theory (1907–1919), commutative algebra (1920–1929), non-commutative algebra and

representation theory (1927–1933), and applications of non-commutative algebra to problems in commutative algebra (1932–1935). [‘Commutative’ here means that the order in which any two elements of the algebra are multiplied has no effect on the result.] … The two major sources of commutative algebra are algebraic geometry and algebraic number theory. Emmy Noether’s two seminal papers of 1921 and 1927 on the subject can be traced, respectively, to these two sources. In these papers, entitled, respectively, ‘Ideal Theory in Rings’ and ‘Abstract Development of Ideal Theory in Algebraic Number Fields and Function Fields,’ she broke fundamentally new ground, originating ‘a new and epoch-making style of thinking in algebra’ (Weyl)” (Kleiner, pp. 91-94).

“[A] fundamental concept which she highlighted in the 1921 paper was that of a ring. This concept, too, did not originate with her. Dedekind (in 1871) introduced it as a subset of the complex numbers closed under addition, subtraction, and multiplication, and called it an ‘order.’ Hilbert, in his famous Report on Number Theory (Zahlbericht) of 1897, coined the term ‘ring,’ but only in the context of rings of integers of algebraic number fields. Fraenkel (in 1914) gave essentially the modern definition of ring, but postulated two extraneous conditions. Noether in her 1921 paper gave the definition in current use” (Kleiner, p. 95).

“As she stated in opening [the offered] paper, ‘the aim of the present work is to translate the factorization theorems of the rational integer numbers and of the ideals in algebraic number fields into ideals of arbitrary integral domains and domains of general rings.’ Drawing on Fraenkel’s formulation of a ring, Noether made the key observation that the abstract notion of ideals in rings could be seen not only to lay at the heart of prior work on factorization in the context of algebraic number fields and of polynomials (in the work of Hilbert, Francis Macaulay (1862-1937), and Emmanuel Lasker (1868-1941)) but also to free that work from its reliance on the underlying field of either real or complex numbers. In her paper she built up that theory – not surprisingly in the context of commutative rings given her motivating examples – in structural terms.

“Noether began by presenting what are essentially the modern axioms for a (commutative) ring … With the notion of a ring thus established, Noether proceeded to define an ideal in terms somewhat different from those employed by Dedekind in 1871. [A subset of a ring is an ideal if the sum and difference of any two elements of the ideal is still in the ideal, and the product of any element of the ideal by any element of the ring is in the ideal.] … Since, for Noether, ideals in general rings were to play the role of factors in the setting of the rational numbers as well as of ideals, in Dedekind’s sense, she needed to define a general notion of divisibility [by ideals] … Another key ingredient to Noether’s approach was the fact that ideals were finitely generated … This assumption allowed her to immediately prove one of the hallmarks of her approach to ring theory, namely, the so-called ascending chain condition on ideals, or, as she termed it, the ‘theorem on finite chains’ … Noether used this structural notion – based on clearly and crisply articulated axioms – to show how to decompose ideals and, in so doing, to establish the factorization theories of her predecessors and contemporaries in the general context of commutative rings” (Katz & Parshall, pp. 441-2).

“Through her ground-breaking papers in which that concept [of a ring] played an essential role, and of which the 1921 paper was an important first, she brought it into prominence as a central notion of algebra. It immediately began to serve as the starting point for much of abstract algebra, taking its rightful place alongside the concepts of group and field, already reasonably well established at that time.

“Noether also began to develop in the 1921 paper a general theory of ideals for commutative rings. Notions of prime, primary, and irreducible ideal, of intersection and product of ideals, of congruence modulo an ideal—in short, much of the machinery of ideal theory, appears here. Toward the end of the paper she defined the concept of module over a non-commutative ring and showed that some of the earlier decomposition results for ideals carry over to submodules …

“The concepts she introduced, the results she obtained, and the mode of thinking she promoted, have become part of our mathematical culture … A number of mathematicians and historians of mathematics have spoken of the ‘algebraization of mathematics’ in the twentieth century. Witness the terminological penetration of algebra into such fields as algebraic geometry, algebraic topology, algebraic number theory, algebraic logic, topological algebra, Banach algebras, von Neumann algebras, Lie groups, and normed rings. Noether’s influence is evident directly in several of these fields and indirectly in others” (Kleiner, pp. 95-100).

“Emmy Noether (1882-1935) had come to mathematics at a time when women were officially debarred from attending university in Germany. Daughter of the noted algebraic geometer and professor at the University of Erlangen, Max Noether (1844-1921), Emmy Noether had grown up among mathematicians and in a university setting but had initially pursued an educational course deemed culturally suitable for one of her sex, that is, she qualified herself to teach modern languages in ‘institutions for the education and instruction of females.’ This vocational preparation aside, Noether studied mathematics – from 1900 to 1902 and then in the winter semester of 1904-1904 – as an auditor and at the discretion of selected professors at her hometown University of Erlangen and at Göttingen University, respectively. At the time, there was no other way for a German woman to attend courses at either institution. Things changed at the University of Erlangen in 1904 and throughout the country four years later, when women were officially allowed to attend and earn degrees from German universities. Noether enrolled at Erlangen in 1904 and earned a doctorate there under the invariant theorist, Paul Gordan, in December of 1907.

“Her degree in hand, however, Noether had nowhere to go within the German system. Even as holders of the doctoral degree, women were allowed neither to hold faculty positions nor even to lecture. Noether thus remained in Erlangen until 1915 as an unofficial assistant to her father, while pursuing her own research agenda. In 1915 and at the invitation of her former professors, Felix Klein and David Hilbert, she finally had the opportunity to make the move to Göttingen. There, she lectured in courses, although in physics, officially listed under Hilbert’s name, beginning in the winter semester of 1916-1917, while Hilbert and Klein fought with the university authorities to allow her to obtain the Habilitation or credentials that would allow her to lecture in her own right. They won that final battle in 1919 in a Germany defeated in World War I and under the new parliamentary representative democratic government of the Weimar Republic that had replaced the old imperial regime” (Katz & Parshall, pp. 438-9).

After the War, “Emmy Noether’s attention shifted to abstract algebra, and by the end of the 1920s she had drawn a group of almost equally gifted young mathematicians around her, Emil Artin and Bartel van der Waerden among them, jocularly called the Noether boys. It is they who turned her way of thinking into ‘Modern Algebra’, while she did fundamental work on commutative and non-commutative algebra.

“In 1933 the Nazis came to power in Germany and set about barring Jews from the civil service and the universities. Emmy Noether was a Jew, and one with well-known left-wing sympathies, but she was able to get a position at the women’s university Bryn Mawr, in Pennsylvania, USA. She died there on 14 April 1935 from a post-operative infection” (Gray, p. 290).

“Noether is best known for her contributions to the development of the then-new field of abstract algebra, as well as ring theory. But one of the reasons Hilbert pushed to bring Noether to Göttingen was the hope that her expertise on invariant theory could be brought to bear on Albert Einstein’s fledgling theory of general relativity, which seemed to violate conservation of energy.

“Noether did not disappoint, devising a theorem that has become a fundamental tool of modern theoretical physics. One of its consequences is that if a physical system behaves the same regardless of its spatial orientation, the system’s angular momentum is conserved. Noether’s theorem applies to any system with a continuous symmetry. When Einstein read Noether’s work on invariants, he wrote to Hilbert: “I’m impressed that such things can be understood in such a general way. The old guard at Göttingen should take some lessons from Miss Noether. She seems to know her stuff” (APS News, vol. 22, no. 3, March 2013). Nobel Laureate Frank Wilczek of MIT has written that Noether’s theorem “has been a guiding star to 20th and 21st century physics” (quoted by Emily Conover, Science News, June 12, 2018).

Gray, A History of Abstract Algebra, 2018. Grolier, Extraordinary Women in Science & Medicine: four centuries of achievement, 2013 (see pp. 85-88). Katz & Parshall, Taming the Unknown, 2014. Kleiner, A History of Abstract Algebra, 2007.

Offprint from: Mathematische Annalen, Band 83, Heft 1/2. 8vo (231 x 158 mm), pp. [1, blank], [24], 25-66. Original printed wrappers, a very fine copy.

Item #4769

Price: \$2,800.00