Just to make a correction: I wrote "Brandy in 1926" instead of "Brandt in 1926"! To say more on this story, my survey article of 1987 writes: ------------------------------------- Brandt???s de???nition of groupoid arose out of his work for over thirteen years [6-10] on generalising to quaternary quadratic forms a composition of binary quadratic forms due to Gauss [63]. Brandt then saw how to use the notion of groupoid in generalising to the non-commutative case the arithmetic of ideals in rings of algebraic integers, replacing the classical ???nite abelian group by a ???nite groupoid [12]. -------------------------------------- [12] is H. BRANDT, ???Idealtheorie in Quaternionenalgebren???, Math. Ann. 99 (1928) 1-29. A review by Baer of Jacobson's 1943 book on "The theory of rings" writes: It is shown that for the two-sided ideals in an order one may obtain unique factorization in the classical sense, including the commutativity of multiplication, and that under comparatively simple necessary conditions on the orders in the ring one may construct Brandt's groupoid of ideals. ------------------------ The 1939 Monograph by A A Albert, who was at Chicago, may also say something but I do not have it to hand. Use of groupoids in this area was surely common knowledge in the 1940s. I also came across published correspondence of Hasse on this topic. Ronnie Brown -- [For admin and other information see: http://www.mta.ca/~cat-dist/ ]