# Section 6.3: Analysis

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 $\mathbb{Z}_p$ and $F[x]/(p(x))$” (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.

###### Theorem 10.31

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 $R\subseteq E\subseteq K$ and E is isomorphic to F.

###### Definition:

An ideal M in a ring R is said to be maximal if $M\neq R$ and whenever J is an ideal such that $M\subseteq J\subseteq R$, then $M=J$ or $J=R$.

Notice the similarities. I’m interested in learning other similar characteristics that they have.