That makes sense. I also posted the question on "yahoo answers" and someone pointed out that "if f is an automorphism, then so is -f and that f+(-f) = 0. 0 is not an not an automorphism so Aut(f) is not closed under point-wise addition of functions. So (Aut(F), +) can not be a group, and so can not be an abelian group. Hence (Aut(F), +, *) is not a ring and so can not be a field."
Does anyone know what the underlying motivation was to develop ring theory? I've heard various arguments stating that ring theory was developed in an attempt to answer questions about the nature of the integers by exploring systems "more inclusive" than the integers themselves. The reason I am asking this is I want to know whether there might be a motiviation to development an abstract theory with three binary operations that satisfy certain distributive laws. Why did mathematicians not develop more abstract theory for two, three, or n associative binary operations on a set? Is it just because no one has gotten around to it, or is it because there is good reason to believe that such theories would not be of much use? Like for example, the complexity of the theory makes the results of work too complicated to be of much practical or theoretical use.
I have at home a book called 'A Modern View of Geometry' (can't recall the author's name just now) in which the properties of projective spaces are gradually developed through a series of finite point models of the axioms. Hand in hand with this, the author sets up an algebraic structure, which functions as a sort of primitive Cartesian geometry. As more axioms are added, the algebra becomes richer and the proto-projective space takes on more of the characteristics we expect from a recognizable space.
The reason I mention this is that the fundamental algebraic object he uses is a TRINARY ring. This is the only place I have seen a trinary ring actually in use as an analytical tool, instead of being the object of the analysis itself.
As a side note, I noticed a very strange coincidence: the 7 point projective space mentioned in the above reference has the same structure as the multiplication table for the octonians, in the sense that each projective line contains exactly 3 points and there is an isomorphic mapping of the unit basis of the octonians to points in the projective space, such that the product of any two elements of the octonian basis (except 1) is precisely the third point on the line containing the two elements that were multiplied.
I don't know what to make of this, but I thought it was an interesting connection between octonians and projective geometry.