Commutative Rings

Abstract

We are at a new beginning. If the goal of Part I was gaining familiarity with algebraic structures, we now wish to understand them at a deeper level. This first chapter of Part II is primarily devoted to commutative rings that are in certain ways similar to the ring of integers. The most prominent example is $F[X]$, the ring of polynomials over a field F. We will see that the theory of divisibility of integers, developed in Section 2.1, holds in essentially the same form not only for $F[X]$, but for more general rings called principal integral domains. In the final part of the chapter, we will establish a theorem on modules over principal ideal domains, along with two surprising applications: one to group theory and the other to matrix theory.