Basics: Groups, Rings, and Fields

In talking about Keith Devlin's call to stop teaching multiplication as repeated addition, I mentioned that multiplication is one of two basic operations defined for a type of algebraic structure called a ring. However, I realized I never explained what that statement means, and in this article, I hope to remedy that by explaining what rings and their related structures (groups and fields) are and why we care about them.

Background concepts

Before getting into groups, rings, and fields, we have to first understand the concepts of sets and binary operations. Mark Chu-Carroll has written a great introduction to set and set operations, but I'll provide a condensed version here.

Intuitively, a set is a well-defined collection of mathematical objects (e.g., numbers, matrices, or functions). We use the term "well-defined" so we can tell whether some arbitrary object is a member or element of a set; the notation "a ∈ S" simply says that object a is an element of set S. A set is non-empty if it contains at least one element.

I should point out that a formalism called axiomatic set theory provides a stricter definition for the term "set" that helps avoid certain paradoxes that would otherwise occur when working with sets. However, for purposes of this article, the simple and intuitive definition of "set" provided above will do.

A binary operation or binary relation on S (represented by a symbol like ◊) is a rule for taking two elements from a set S (call them a and b) and combining them to get a third object (call it c); we write this as "c = a ◊ b". Notice that there is no intrinsic requirement for c to be an element of S. However, if combining any pair of elements of S using the operation ◊ always results in an object that is also an element of S, then we say that "S is closed under ◊". Mathematically, we say "S is closed under ◊ if and only if for all a ∈ S and b ∈ S, a ◊ b ∈ S."

A algebraic structure consists of one or more sets closed under one or more binary operations that satisfy certain conditions.

Defining groups, rings, and fields

One of the simplest algebraic structures is a group, which consists of a non-empty set S and a binary relation on S (written as ◊) that satisfies the following conditions:

1. Closure under ◊: For all a, b ∈ S, a ◊ b ∈ S.
2. Associativity of ◊: For all a, b, c ∈ S, (a ◊ b) ◊ c = a ◊ (b ◊ c); i.e., the order in which we apply ◊ doesn't matter.
3. Existence of an identity element: There is some element e ∈ S such that for all a ∈ S, a ◊ e = e ◊ a = a.
4. Existence of inverse elements: For every element a ∈ S, there is some element a^-1 ∈ S such that a ◊ a^-1 = a^-1 ◊ a = e. The element a^-1 is called the inverse of a.

A group defined in this way is written using the notation (S, ◊).

There are two things to notice. First, the definition of a group requires us to provide or define the operation ◊; in other words, saying "Set S is a group" without saying anything about some binary relation on S is a meaningless statement. Second, the definition of a group does not require ◊ to be commutative; that is, we don't require that a ◊ b = b ◊ a for all a, b ∈ S. If ◊ does commute, then the group (S, ◊) is called a commutative, or Abelian, group.

A familiar example of a group is the set of integers Z = {…, -2, -1, 0, 1, 2, …} under ordinary addition (i.e., the addition taught in elementary school). For this group, the identity element is 0, and the inverse element of a given integer, a, is the negative of that integer, −a.

Like a group, a ring is a type of algebraic structure, but defined using two binary operations. The ring (S, ⊕, ⊗) consists of a non-empty set S, a "ring addition" relation ⊕, and a "ring multiplication" relation ⊗ and satisfies the following conditions:

1. Closure under ⊕: For all a, b ∈ S, a ⊕ b ∈ S.
2. Closure under ⊗: For all a, b ∈ S, a ⊗ b ∈ S.
3. Commutativity of ⊕: For all a, b ∈ S, a ⊕ b = b ⊕ a.
4. Associativity of ⊕: For all a, b, c ∈ S, (a ⊕ b) ⊕ c = a ⊕ (b ⊕ c).
5. Existence of an identity element for ⊕: There is some element z ∈ S such that for all a ∈ S, a ⊕ z = z ⊕ a = a. The element z is called the additive identity or the zero element of the ring.
6. Existence of inverse elements for ⊕: For every element a ∈ S, there is some element x ∈ S such that a ⊕ x = x ⊕ a = z. The element x is called the additive inverse of a.
7. Associativity of ⊗: For all a, b, c ∈ S, (a ⊗ b) ⊗ c = a ⊗ (b ⊗ c).
8. Distributive law of ⊗ over ⊕: For all a, b, c ∈ S, a ⊗ (b ⊕ c) = (a ⊗ b) ⊕ (a ⊗ c) and (b ⊕ c) ⊗ a = (b ⊗ a) ⊕ (c ⊗ a).

As noted above with the definition of a group, saying "S is a ring" without defining both the ⊕ and ⊗ relations is a meaningless statement. Also, notice that the ring multiplication relation ⊗ need not be commutative; because of this, the order in which we write the terms in the distributive law (#8) above matters. If ⊗ is commutative, then the ring is called a commutative ring.

A field is a ring that satisfies two additional conditions:

1. Existence of an identity element for ⊗: There is some element u ∈ S, u ≠ z, such that for all a ∈ S, a ⊗ u = u ⊗ a = a. The element u is called the multiplicative identity or the unity element of the ring. Note that the multiplicative identity and the additive identity cannot be the same element of S.
2. Existence of inverse elements for ⊗: For every element a ∈ S, a ≠ z, there is some element a^-1 ∈ S such that a ⊗ a^-1 = a^-1 ⊗ a = u. The element a^-1 is called the multiplicative inverse of a.

The set of integers Z under ordinary addition and ordinary multiplication, written (Z, +, ×), form a ring. In this ring, the integers a and −a are additive inverses of each other, 0 is the additive identity, and 1 is the multiplicative identity. However, (Z, +, ×) is not a field because, except for the case a = 1, the multiplicative inverse of a ∈ Z is not a member of Z; e.g., the multiplicative inverse of 2 is 1/2, which is not an integer.

The set of real numbers R under ordinary addition and ordinary multiplication is a field (and thus a ring as well). The additive inverses, additive identity, and multiplicative identity are the "same" as those for integers. However, unlike the integers, the multiplicative inverse r^-1 of every non-zero real number r ∈ R is itself a member of R.

What Devlin was talking about

Looking at the definition of a ring above, you can see (fairly easily, I hope) what Devlin meant when he said that multiplication is not repeated addition, but is instead another "basic operation" you can perform on numbers. Nothing in the definition of a ring requires us to assert that ring multiplication is repeated ring addition; in fact, the definition of a ring requires no relation between multiplication to addition, except through the distributive law.

Devlin's arguments boil down to this: Because rings are typically the structures through which mathematicians understand the numbers we commonly deal with (e.g., the integers, rationals, reals, and complex numbers) and because of the way a ring is defined, teachers should not introduce a concept (multiplication = repeated addition) that really isn't there.

Why care about groups, rings, and fields?

From a pure math perspective, proving statements using only the definitions above automatically tells us the properties of a large range of mathematical objects of interest. For example, using only the definition of a field, we can show that the additive and multiplicative identities must be unique, as must the pairs of additive and multiplicative inverses (i.e., each element of a field has a unique additive and a unique multiplicative inverse). Thus, even though the rules for adding and multiplying numbers, matrices, and functions differ, we know that so long as these sets and their operations form a field, the uniqueness properties of identities and inverses must hold. Without such broad theories, every time a new field is defined, we would have to go back and prove these uniqueness criteria using a different set of rules.

Moreover, the theory underlying these structures has played a key role in the mathematical development of modern science, ranging from physics and chemistry (which rely on group theory to understand the symmetries found in nature) to computer science and electrical engineering (which rely on group theory to develop the coding systems used in encryption and digital communication systems). Without these algebraic structures, many of the results that present day scientists and engineers depend on would have been difficult, if not impossible, to derive and understand.