Ars Mathematica

Well, okay, not really lies, but I formed ideas in my abstract algebra class that I later had to unlearn:

• Most integral domains are unique factorization domains. In reality, integral domains are almost never UFDs. Beyond the examples usually taught, there is one additional large class of UFDs, regular local rings, and then just scattered examples. If you take a polynomial ring in two or more variables, and mod out by any random prime ideal, you will almost certainly not get a UFD. For example, in the ring k[x,y]/(x^2+y^2-1), x^2 also factors as (1-y)(1+y)
• Abelian groups are a direct sum of cyclic groups. While this is true for finitely-generated abelian groups, it is far from true for infinitely-generated abelian groups, even if you consider infinite direct sums. A typical infinitely-generated abelian group is Q, which cannot be written as a direct sum of any subgroups, but very much not cyclic.
• Finite-dimensional noncommutative division rings over the rationals are all subrings of the quaternions. The definition of the quaternions makes sense with coefficients in any subfield of the reals, and gives you a finite-dimensional division ring over that subfield. If the subfield is itself finite-dimensional over Q, this gives you a finite-dimensional division algebra over Q. I thought that this construction gave you all of the possibilities. This is far from the case. The quaternions are 4-dimensional over their center, but you can construct other division algebras of any dimension over their center, as long as that dimension is a perfect square.

Did this happen to anyone else?