This section on “The Structure of R/I when I is Prime or Maximal” provided a lot of information in very small number of pages. It defined what it means for a commutative ring to be prime and what a maximal is. In defining the former Hungerford explains “[q]uotient rings were developed as a natural generalization of the rings and ” (page 162). It is relieving to have this comparison explicitly stated, albeit a section later.
Maximals reminds me of subfields and fields of quotients. Their definitions are very very similar.
Let R be an integral domain and F its field of quotients. If K is a field containing R, then K contains a field E such that and E is isomorphic to F.
An ideal M in a ring R is said to be maximal if and whenever J is an ideal such that , then or .
Notice the similarities. I’m interested in learning other similar characteristics that they have.