So in a PID, whether Euclidean or not, the always exists. However, it is only in a ED that computing s is easily possible via the Euclidean Algorithm. Then from the previous example we have . Back-substitution gives , so finally . The next step is to show that every PID is also a unique factorization domain.
Every prime is irreducible.
In a PID, every irreducible is prime. The last property will be crucial in proving the uniqueness of factorizations into irreducibles, but for the existence we need to do some more preparation. Let be a sequence of elements of R with . Then there exists k such that , so the chain of ideals stabilizes. Hence any strictly ascending chain of ideals must be finite.