The first three chapters of this text tell the story of unique factorization. The culmination is the result that any Euclidean domain is a unique factorization domain; that is, in an integral domain with a well-behaved division algorithm, a nonzero nonunit necessarily factors uniquely into irreducibles. In order to expediently develop that result, we ignored many concepts that are otherwise interesting and useful in a first course in abstract algebra. This chapter is a coda that seeks to fill in some of those gaps.
In Section 4.1, we expand on the definition of ideal introduced in Section 2.4 and explore non-principal ideals. No math course is complete without a discussion of functions of some sort; we explore homomorphisms in Section 4.2 Finally, in Section 4.3, we introduce prime and maximal ideals, as well as the notion of congruence modulo \(I\) and use ideals to build new rings from old. We conclude with an exploration and proof of the First Isomorphism Theorem.
