Whenever I have seen unique factorization discussed, it is always with respect to the solution of diophantine equations; the equations are solved by splitting the equation into linear functions over a ring and then invoking unique factorization. But the discussions always give the impression that the failure of unique factorization causes all sorts of problems. Apart from the applications to diophantine equations, why is unique factorization such a desireable property?

Whenever I have seen unique factorization discussed, it is always with respect to the solution of diophantine equations; the equations are solved by splitting the equation into linear functions over a ring and then invoking unique factorization. But the discussions always give the impression that the failure of unique factorization causes all sorts of problems. Apart from the applications to diophantine equations, why is unique factorization such a desirable property?