Prime and Maximal Ideals in a PID (Algebra 2: Lecture 5 Video 1)

Lecture 5: We started this lecture by recalling what it means for an ideal to be maximal or to be prime and also some important results from Section 7.4.  We then proved that in a PID every prime ideal is maximal.  Using this result we proved that if R[x] is a PID, then R is a field.  We defined what it means for an element to be irreducible and what it means for an element to be prime. We showed that in any integral domain prime elements are irreducible.  We proved that in a PID irreducible elements are also prime.  We gave an example of a ring, Z[sqrt{-5}], which has an element that is irreducible but not prime.  In the last video we gave an example (without the details of the argument) of a ring that is a PID but not a Euclidean domain.

Reading: We started this lecture by recalling some important material from Section 7.4 of Dummit and Foote.  We then closely followed the presentation of Section 8.2 when proving Proposition 7 and Corollary 8.  We then jumped ahead into Section 8.3 where we gave the definitions of irreducible, prime, and associate on page 284.  We also gave the proofs of Propositions 10 and 11 in that section.  In the last video, we returned to the end of Section 8.1 to show that Z[(1+sqrt{-19})/2] is not a Euclidean domain.  We then gave a sketch of the argument proving that this ring is a PID that came from the end of Section 8.2.