Consider the following page from Noether and Schmeidler's 1921 work:
And pay special attention to note 10 on page 7. It is not clear to me whether the principal ideals (Hauptideale) were the focus of Hurwitz's work as opposed to Noether and Schmeidler's, or, as I guess,the other way round. Whichever the case, could you illustrate the differences and explain the role of the quaternions in the whole discussion?
Thanks in advance,