Emmy Noether and the development of commutative algebra. (English) Zbl 0930.13001
Teicher, Mina (ed.), The heritage of Emmy Noether. Proceedings of the conference, Bar-Ilan University, Ramat-Gan, Israel, December 2–3, 1996. Ramat-Gan: Bar-Ilan Univ., The Emmy Noether Research Institute of Mathematics, Isr. Math. Conf. Proc. 12, 23-38 (1999).
This article gives a nice concrete description of the role of E. Noether in the development of commutative algebra. It starts with a description of the roots of algebra in number theory and algebraic geometry. These roots include the work of D. Hilbert and E. Kummer on the ideal theory of algebraic integers, as well as Hilberts fundamental work on invariant theory. The efforts of R. Dedekind and Weber to give an algebraic foundation of the theory of Riemann surfaces, and in particular of the Riemann-Roch theorem, were important, as were the contributions of L. Kronecker to algebra and geometry. Particularly the influence on E. Noether of the ideas of Dedekind is emphasized. Finally the contribution of E. Lasker to ideal theory, E. Steinitz’ fundamental work on fields, and F. S. Macaulay’s modular systems are mentioned.
Noether’s contributions are treated chronologically. First her work on invariant theory is mentioned. Although she was considered one of the leading experts in the field it is not clear exactly what her contributions were. A detailed description is given of her contributions to commutative algebra. The table of contents of two of her most fundamental articles in Math. Ann. [E. Noether, Math. Ann. 83, 24-66 (1921; JFM 48.0121.03) and 96, 26-61 (1926; JFM 52.0130.01)] is presented. It is pointed out how her systematic use of chain conditions allowed her to avoid eliminiation theory, which was a standard technique in algebra. Her decomposition theory is described, and the author gives a vivid description of her new ideas and the advantage of the axiomatic treatment that she used.
It is amazing that so much of the algebraic material that today is standard in any texbook in algebra was introduced by E. Noether, or profoundly influenced by her. The axiomatic treatment, ideal theory, the chain conditions, integral dependence, isomorphism theorems, and the relation between the Krull dimension and the transcendence degree of algebras, are all illustrative examples. A short account of her contributions to non-commutative algebra is given.
For the entire collection see [Zbl 0906.00012].
Noether’s contributions are treated chronologically. First her work on invariant theory is mentioned. Although she was considered one of the leading experts in the field it is not clear exactly what her contributions were. A detailed description is given of her contributions to commutative algebra. The table of contents of two of her most fundamental articles in Math. Ann. [E. Noether, Math. Ann. 83, 24-66 (1921; JFM 48.0121.03) and 96, 26-61 (1926; JFM 52.0130.01)] is presented. It is pointed out how her systematic use of chain conditions allowed her to avoid eliminiation theory, which was a standard technique in algebra. Her decomposition theory is described, and the author gives a vivid description of her new ideas and the advantage of the axiomatic treatment that she used.
It is amazing that so much of the algebraic material that today is standard in any texbook in algebra was introduced by E. Noether, or profoundly influenced by her. The axiomatic treatment, ideal theory, the chain conditions, integral dependence, isomorphism theorems, and the relation between the Krull dimension and the transcendence degree of algebras, are all illustrative examples. A short account of her contributions to non-commutative algebra is given.
For the entire collection see [Zbl 0906.00012].
Reviewer: Dan Laksov (Stockholm)
MSC:
| 13-03 | History of commutative algebra |
| 01A60 | History of mathematics in the 20th century |
| 13Exx | Chain conditions, finiteness conditions in commutative ring theory |
| 16-03 | History of associative rings and algebras |
