section_path: University of Göttingen/Work in abstract algebra
section_name: Work in abstract algebra
paragraph_text: Work in abstract algebra
  
  Although Noether's theorem had a significant effect upon classical and quantum mechanics, among mathematicians she is best remembered for her contributions to abstract algebra. In his introduction to Noether's Collected Papers, Nathan Jacobson wrote that
  
  The development of abstract algebra, which is one of the most distinctive innovations of twentieth century mathematics, is largely due to her – in published papers, in lectures, and in personal influence on her contemporaries.
  
  Noether's work in algebra began in 1920 when, in collaboration with her protégé Werner Schmeidler, she published a paper about the theory of ideals in which they defined left and right ideals in a ring.
  
  The following year she published the paper Idealtheorie in Ringbereichen, analyzing ascending chain conditions with regards to (mathematical) ideals, in which she proved the Lasker-Noether theorem in its full generality. Noted algebraist Irving Kaplansky called this work "revolutionary". The publication gave rise to the term Noetherian for objects which satisfy the ascending chain condition.
  
  In 1924, a young Dutch mathematician, Bartel Leendert van der Waerden, arrived at the University of Göttingen. He immediately began working with Noether, who provided invaluable methods of abstract conceptualization. Van der Waerden later said that her originality was "absolute beyond comparison". After returning to Amsterdam, he wrote Moderne Algebra, a central two-volume text in the field; its second volume, published in 1931, borrowed heavily from Noether's work. Although Noether did not seek recognition, he included as a note in the seventh edition "based in part on lectures by E. Artin and E. Noether". Beginning in 1927, Noether worked closely with Emil Artin, Richard Brauer and Helmut Hasse on noncommutative algebras.
  
  Van der Waerden's visit was part of a convergence of mathematicians from all over the world to Göttingen, which had become a major hub of mathematical and physical research. Russian mathematicians Pavel Alexandrov and Pavel Urysohn were the first of several in 1923. Between 1926 and 1930, Alexandrov regularly lectured at the university, and he and Noether became good friends. He began referring to her as der Noether, using the masculine German article as a term of endearment to show his respect. She tried to arrange for him to obtain a position at Göttingen as a regular professor, but was able only to help him secure a scholarship to Princeton University for the 1927–1928 academic year from the Rockefeller Foundation.