What was the evolution of "basis" and "generating set" in algebra?

Question

Today, I've heard someone speak of a basis (of an ideal), meaning a generating set. All the time, I was fine with the term Gröbner-basis, but when it comes without the prefix, it's a bit funny, since basis, morally, is earmarked for something generating freely.

So I wondered which term was used first, and how, as well as by whom of course. The first thing that came to my mind was Hilbert's basis theorem, but Hilbert did not speak about bases. The next names to consider were Gröbner and Buchberger. And in fact, Gröbner used Basis for generating systems of ideals whereas speaking about a module, a basis was supposed to generate freely. (See, e.g., Moderne Algebraische Geometrie, 1949, Springer Wien & Innsbruck.) Not surprisingly, being a student of Gröbner, Buchberger also called generating sets of ideals bases.

I realise that a full description of the evolution of basis could be too much to ask for, so I would already be happy to read an answer which builds a bridge between Hilbert's basis theorem (Über die Theorie der algebraischen Formen, 1890) and Gröbner's terminology, possibly by pointing at the first one to refer to it as basis theorem.

Answer 1

I do not have conclusive evidence, but I will name likely suspects and make a circumstantial case. Hopefully, it will get the ball rolling.

"Basis" as generating set of ideals may have been introduced before Hilbert by Dedekind, even if Hilbert did not use the word. Dedekind introduced "ideals" in (what we call) number rings only in the supplements to 1879 edition of Vorlesungen über Zahlentheorie, but used "modul" back in 1871. A generating set of the ideal is then naturally a basis of the "modul" by analogy to vector space basis. In 1882 Dedekind and Weber "related geometric ideas with rings of polynomials and extended the use of modules" according to MacTutor's Ring Theory. By the way, while "ideal", "module" and "field" are Dedekind's neologisms, "ring" is Hilbert's from 1893 (in print 1897).

What we know for sure is that van der Waerden in his celebrated Modern Algebra uses "basis" as a matter of course, and explicitly relates that to the module interpretation. He also calls Hilbert basis theorem by this name, and is most likely Gröbner's and Buchberger's source.

The two volumes of Modern Algebra were published in 1930 and 1931, but written in 1926-1929 under the heavy influence, by van der Waerden's own admission, of Emmy Noether. Van der Waerden became the modern algebraist when working under Noether in Göttingen in 1924-1925. Noether herself was converted to the Dedekind-Hilbert methods by Fischer back in Erlangen since 1911, and quickly became such an authority on them that Hilbert and Klein invited her to Göttingen in 1915. Her seminal paper Idealtheorie in Ringbereichen (1921) is the seed of Modern Algebra, and gives the decomposition of ideals into intersections of primary ideals in commutative rings with ascending chain condition. Now these are called "Noetherian rings", and the Hilbert "basis" theorem is proved for polynomial rings over them. My guess is that van der Waerden got his usage of "basis" from Noether, and that she is the one who christened the Hilbert's theorem by its modern name. She might also have been the first to use "basis" this way, but it could be Dedekind or somebody in between.