Advanced Mathematics

6.5Rings and Fields

Purpose of Section

To introduce the concept of an algebraic ring and an important type of ring called a field.

6.5.1 Introduction to Rings

Although the algebraic group has one binary operation, the algebraic system we studied in grade school has two binary operations: addition and multiplication. This leads us to the study of rings and fields. We begin with one of the most important abstract systems with two binary operations called a ring.¹ A ring is one of the basic structures of abstract algebra, which generalizes the common arithmetic operations of the integers, polynomials, matrices, and so on. Ring theory is used today to understand basic physical laws, such as those underlying such things as symmetry phenomena in molecular chemistry.

You have seen examples of rings before. The integers ℤ with ordinary addition (+) and multiplication (×) are an example of an algebraic ring. In this regard, you might think of a ring as “generalized integers.” The study of rings was initiated (in part) by the German mathematician Richard Dedekind (1831–1916) in the late 1800s, and the axiomatic foundations were laid down in the 1920s, and most of them by the German mathematician Emmy Noether.

Definition

A set {R, +, ×} with two (closed) binary operations of +(addition) and ×(multiplication) is called a ring if:

1. The system with operation + forms a commutative group.
2. The operation × is associative. In the language of predicate logic:
3. The operation × distributive ...