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.

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