Research On The Early History Of Commutative Ring Theory

As an important branch of mathematics, abstract algebra mainly studies mathematical structures of group, ring, field, module, lattice and so on. Ring theory is one of the deepest sub-disciplines in abstract algebra. According to whether multiplication is commutative or non-commutative, ring is branched to commutative ring and non-commutative ring. Although both commutative ring theory and non-commutative ring theory arose from the early 19th century, their origins and development path were quite different. Commutative ring theory originated from algebraic number theory, algebraic geometry as weel as invariant theory. Among them, algebraic number theory was the most vital root for its development. In turn, commutative ring theory also mainly applied to those areas. Due to the efforts of many mathematicians, including Gauss, Dedekind, Kronecker, Hilbert, Fraenkel, Emmy Noether and others commutative ring theory came to mature and had stimulated lots of applications in the 1920-30s.This dissertation is based upon a large number of original documents and research results in the 19th and 20th centuries. By using the concept analyzing method to explore specific scientific practice and theoretical background of the origin, development, perfection and dissemination of commutative ring theory from the 19th century through the 1920s, its development context and evolution trend are summarized; by using the comparative study method, the characteristics and patterns of commutative ring theory are studied, and the key figures’different research methods and results to deal with the concepts and theories of commutative ring theory are analyzed; by using a comprehensive application of the empirical methods and chronicle methods of historical materials, the overall history of commutative ring theory is clarified and some reasonable historical evaluations are given. Finally a panoramic scene of commutative ring theory was formed, which in turn elaborated the evolution of commutative ring theory, the connections between the theory and the other disciplines of mathematics, as well as ideological heritage among all academics. This had important both theoretical and practical significances to the understanding and the awareness of commutative ring theory, ring theory and relevant disciplines. The research results and conclusions are:(1) Based upon the core problems of classical number theory, such as the higher reciprocity laws, quadratic form in two variables and Fermat’s Last Theorem, this dissertation focuses on their key problem of unique factorization and studies the origin of the commutative ring theory in algebraic number theory. The conclusion shows that Gauss, Kummer, Dedekind, Kronecker, Hilbert and other mathematicians played important roles in the historical process. Some concepts, e.g. complex integral ring, ideal number, ideal, order ring, ring, and etc., gradually became clearer and clearer. This not only laid the foundation for one dimensional commutative algebra, but also established the discipline of algebraic number theory.(2) The historical process, in particular the basis theorem and the Nullensatz Theorem proved by Hilbert, as well as the primary ideal and the primary decomposition theory of Lasker and Macaulay, of coming up of commutative ring theory in algebraic geometry and invariant theory is revealed.(3) It is reconfirmed that Fraenkel was the first mathematician who proposed the concept of abstract ring and it is shown how Fraenkel followed the academic guidance and helps from such great men as Loewy, Hensel, Hilbert, Steinitz and Zermelo to move forward to a correct innovational road and to study the theory of commutative ring with the belief in the axiomatic thought. The conclusion is that Fraenkel applied the axiomatic approach from concrete mathematical examples and brought the new mathematical axiomatic thought to a higher theoretical level, therefore paved the way for its further development. Axiomatization and abstraction were his central idea to engage in mathematical research, and was also an important factor for his great success in the study of the axiomatic set theory.(4) The reason why Emmy Noether turned her study from invariant theory to commutative ring theory is answered. The dissertation reveals that Emmy Noether completed her axiomatical characterizaton for abstract ring, especially for Noetherian ring, with the help of emphasis on and application of the ascending chain condition, and formally established the abstract theory of commutative ring, and thus promoted the subject of abstract algebra mature.(5) In the aspect of the origin and development of non-commutative ring theory, this dissertation expounds that it was derived from the generalization of complex numbers to a variety of hypercomplex systems. This study is vital to understand the status of commutative ring theory in the whole ring theory.(6) The relationships, between ring theory and group theory, field theory, algebraic geometry, modular category, physics and lattice theory etc. are discussed. The conclusion is that, together with its applications at the beginning, commutative ring theory was born with its application. While ring theory was being established to a relative complete state, its gradual increased theoretical level broadened its application range even further, and penetrated into various branches of mathematics, and built up close and natural relationship with these branches. The mutual penetrations and influences between each other will be a major developing trend in the future.(7) The history of commutative ring theory was magnificent. It had links to the famous historical problems such as Fermat’s Last Theorem, the higher reciprocity laws and had close connection with multiple disciplines of algebraic number theory, algebraic geometry and invariant theory. At the same time, it was an epitome for the concept of algebra from number to set, and to structure from the 19th century through the 1920s, in which axiomatization and structuring occupied a dominant position.(8) The ideological heritage among mathematicians in the field of commutative ring theory was exciting and profound and was smoothly carried on by generations as well. It condensed the mathematical thought essence from a number of masters, such as Gauss, Kummer, Dedekind, Kronecker, Lasker, Macaulay, Hilbert, Fraenkel, Emmy Noether, Artin, Van der Waerden, Zeng Jiong, and so on. It was a glorious chapter in the history of modern mathematics, which not only greatly promoted the evolution of modern mathematics, but also had the spiritual benefit to the whole mankind in the field of humanities.(9) Because of Zeng Jiong’s close student-teacher relation with Emmy Neother and Artin, and in view of the place and historical period that Zeng Jiong once lived and/or worked, the relevant contents are studied (please see appendix).