Every PID is a UFD (Algebra 2: Lecture 6 Video 5)

Lecture 6: We started this lecture by talking about hw a nonzero nonunit integer can be written as a product of primes and their negatives. We then stated that the analogous statement holds in any PID. That is, every nonzero nonunit element factors into a finite product of irreducible elements. In order to prove this we defined what it means for a ring to satisfy the ascending chain condition (ACC) on ideals. We proved that a commutative ring satisfies the ACC on ideals if and only if each of its ideals is finitely generated. We used the fact that a PID satisfies the ACC on ideals to prove the theorem stated above. We split this into two major parts, first showing that a nonzero nonunit element had to have some irreducible factor. We then defined a Unique Factorization Domain (UFD) and gave several examples. We also gave some non-examples, showing that  and  are not UFDs. We then proved the analogue of a result we proved in the last lecture, showing that in a UFD every irreducible element is prime. We proved a result showing that if we are given elements in a UFD that are already factored into products of irreducible elements, it is easy to write down a greatest common divisor for them. Finally, we proved that every PID is a UFD. The key idea here was to use induction on the number of irreducible elements occurring in a factorization.

Reading: In this lecture we covered everything in Section 8.3 that was not covered in the previous lecture except the section at the end about Factoring in the Gaussian Integers. But, we did not cover everything in the same order. The statement that in a PID every nonzero nonunit element can be written as a finite product of irreducible elements is part of the proof that every PID is a UFD. Instead of just showing that a PID satisfies the ACC on ideals, we proved something more general, which is essentially part of Theorem 1 in Section 12.1 (but that result is stated in the language of modules, so it might be challenging to understand right now). We then gave the definition of UFD and several examples from pages 285-286. We proved Propositions 12 and 13 and finished the lecture by giving the rest of the proof of Theorem 14 (the uniqueness part).