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.

###### Theorem 10.31

Let

Rbe an integral domain andFits field of quotients. IfKis a field containingR, thenKcontains a fieldEsuch that andEis isomorphic toF.

###### Definition:

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.