Histoire Des Mathématiques

T he theory of rings has deep historical roots in several of the mainstream, classical disciplines of nineteenth-century mathematics, such as the theory of algebraic number fields, the theory of algebraic functions, the theory of polynomial forms, the theory of quaternions and hypercomplex numbers, and others. The unification of all these trends reached its peak with the works of Emmy Noether (1882-1935) and Wolfgang Krull (18991970). Using the abstract concept of ring, these two mathematicians published systematic and comprehensive studies of theorems — especially factorization theorems — common to all these domains. The first attempt to analyze the concept of abstract ring itself, as an autonomous mathematical entity, was published by Abraham Halevy Fraenkel (1891-1965) in 1914. However, as many historical accounts justly care to stress, Fraenkel’s work in this direction had a rather marginal influence on the development of the theory. Fraenkel’s work was not organically connected to any of the main streams that played a major role in the development of the theory of rings and it did not directly contribute to change their courses. What was, then, the historical framework within which Fraenkel’s work arose ? The present article sketches the answer to this question, by tracing the roots of Fraenkel’s ideas back to three main sources : the works of the American school of postulational analysis, Kurt Hensel’s theory of p-adic numbers and Ernst Steinitz’s work on abstract fields. By doing this, the actual historical place of Fraenkel’s work and its actual significance for the development of the theory will become clearer. At the same time an additional historical point will be stressed, namely, that the idea of an abstract ring and that of an ideal — two ideas so closely connected in the modern conception of algebra — arose in two separate paths that were only connected in the work of Emmy Noether.