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.

The first thing one notes about a ring is that practically no conditions are imposed on the multiplication, only that it is associative and obeys distributive laws. The action in a ring lies with the addition.

We denote the ring operation of addition by + and multiplication by ×, although they do not necessarily denote addition and multiplication of numbers. We often denote ring multiplication by ab for shorthand. We also call the additive identity in the ring, the zero (or additive identity) of the ring, and denote it by 0. A ring need not have a multiplicative identity, but when it does, we say the ring has a multiplicative identity (or unity) and is denoted by, you guessed it, 1.

6.5.2 Common Rings

• Example 1 The integers ℤ = {0, ± 1, ± 2, …} with usual addition and multiplication are a commutative ring with the multiplicative identity of 1.
• Example 2 The set ℤ[x] of all polynomials in x with integer coefficients and usual addition and multiplication is a commutative ring with multiplicative identity f(x) = 1.
• Example 3 The set 2ℤ = {0, ± 2, ± 4, …} of even integers with usual addition and multiplication is a commutative ring without a multiplicative identity.
• Example 4 The set ℤ_n = {0, 1, 2, … , n − 1} with addition and multiplication modulo n is a commutative ring with multiplicative identity 1. This ring is called the ring of integers modulo n.
• Example 5 The set M₂(ℤ) of all 2 × 2 matrices with integer entries is a noncommutative ring with multiplicative identity
  
• Example 6 The sets ℤ, ℚ, ℝ with the usual addition and multiplication are all rings. The additive identity in each of these rings is 0 and the multiplicative identity is 1.
• Example 7 An important ring is the set of polynomials of the form
  

  for a given natural number n and real coefficients with normal addition and multiplication of polynomials. Note that addition, subtraction, and multiplication are closed operations, but division is not.

Note that the addition table forms a commutative group, and if we only look at the nonzero members {1, 2} of the multiplication table, they form a commutative group of order two. This is special kind of ring called a field. But not all rings have this much structure, as the following example illustrates.

Although the addition table for ℤ₆ forms a commutative group with additive identity 0, the nonzero members of the multiplication table do not form a group for a variety of reasons. First off, the numbers 2, 3, and 4 do not have multiplicative inverses since there is no member of the ring that satisfies the equations 2 × a = 1, 3 × a = 1, 4 × a = 1.

Although rings are important algebraic structures in many areas of mathematics,² they sometimes are too restrictive. For example, in the ring of integers ℤ under ordinary addition and multiplication, we cannot divide 3 by 5 to obtain a member of the ring. This leads us to the study of the algebraic field.³

6.5.3 Algebraic Fields

Rings allow for addition, subtraction, and multiplication, but not division. The equation 3x = 7 has no solution in the ring of integers but does have a solution in the field of rational numbers. An algebraic field is the domain for many areas of mathematics, including real and complex analysis. We saw in Section 4.2 that the real numbers were a complete ordered field.

When one thinks of a field, one thinks of a structure with two operations resembling addition and multiplication, where one can add, subtract, multiply, and divide. The reader is well aware of three common fields from analysis, the rational numbers ℚ, the real numbers ℝ, and the complex numbers ℂ. There are “hypercomplex” number systems past the complex numbers, like four‐dimensional quaternions and the eight‐dimensional octonians, neither of those number systems are fields, but more general abstract systems. Quaternion multiplication does not commute, and octonian multiplication not only does not commute but also is not even associative.

6.5.4 Finite Fields

Although we think of the rational numbers ℚ, the real numbers ℝ, and the complex numbers ℂ as the major infinite fields, there are many important finite fields, which contain only a finite number of elements. Finite fields play an important role in many areas of mathematics, including algebraic geometry and number theory, as well as applied areas like coding theory and cryptography.

An interesting property of finite fields is that they exist only for certain orders. For example there are finite fields of order 2, 3, 4, and 5, but none of order 6. There are finite fields of order 7, 8, 9, but none of order 10. To be specific, there are finite fields of order pⁿ, n = 1, 2, … where p is a prime number, but no others. These finite fields are called Galois fields and denoted by GF(pⁿ).⁴

There are two main classifications of finite fields. There are the finite fields GF(p) of prime order and then the more involved fields of order pⁿ when n > 1. The prime order fields GF(p), when n = 1, are the field of permutations ℤ_p = {0, 1, 2, … , p − 1} of integers with addition and multiplication carried out modulo p. However, we saw in Example 8 that ℤ₆ = {0, 1, 2, 3, 4, 5} is not a field since 6 is not a prime number.