Give an example of a ring where unique factorization fails, and prove the failure.
Solution
Take \[ R=\mathbb Z[\sqrt{-5}]. \] Its norm \[ N(a+b\sqrt{-5})=a^2+5b^2 \] is multiplicative. The only units are \(\pm1\), since \(N(u)=1\) for a unit \(u\).
In \(R\), \[ 6=2\cdot3=(1+\sqrt{-5})(1-\sqrt{-5}). \] There is no element of norm \(2\) or \(3\): if \(b\ne0\), then \(a^2+5b^2\ge5\), while for \(b=0\) the norm is a square. Hence:
- \(2\) is irreducible, because a nontrivial factorization would force norms \(2\) and \(2\);
- \(3\) is irreducible, because a nontrivial factorization would force a factor of norm \(3\);
- \(1\pm\sqrt{-5}\) are irreducible, because a nontrivial factorization of an element of norm \(6\) would require a factor of norm \(2\) or \(3\).
The two factorizations are not associate: the factors \(2,3\) have norms \(4,9\), whereas \(1\pm\sqrt{-5}\) have norm \(6\), and associates have equal norm. Thus \(6\) has two inequivalent factorizations into irreducibles, so \(R\) is not a UFD.