Integral Domain: Definition, Examples, Properties

A commutative non-trivial ring with unity and no zero divisors is called an integral domain. For example, the set Z of integers is an integral domain. In this article, let us study integral domain along with its definition, examples, and a few solved problems.

Definition of an Integral Domain

Let R be a non-trivial ring with unity. The ring R is said to be an integral domain if it is commutative and does not have zero divisors.

For example, the ring (ℝ, +, ⋅) is an integral domain as it is a commutative ring with unity 1 and contains no zero divisors.

Examples of Integral Domains

A list of few examples of integral domains are given below.

1. The rings (ℚ, +, ⋅), (ℤ, +, ⋅) are commutative having no zero divisors, so they are integral domains.
2. The ring (ℤ_n, +, ⋅) of integers modulo n is an integral domain if n is a prime number. So (ℤ₅, +, ⋅) is an integral domain.
3. The ring ℤ[i] = {a+bi : a, b ∈ ℤ} of Gaussian integers is an integral domain. Similarly, ℤ[√2] = {a+b√2 : a, b ∈ ℤ}, ℤ[√3] are integral domains.
4. The ring ℤ[x] of polynomials with integer coefficients is an integral domain.

Non-Examples of Integral Domains

There are many rings which are not integral domains. A list of such rings are given below.

1. The zero ring is not an integral domain.
2. The ring Z₆ is not an integral domain although it is commutative. This is because Z₆ contains zero divisors as $\bar{2} \cdot \bar{3}=\bar{0}$.
3. The ring 2Z is not an integral domain although it is a commutative ring with no zero divisors. Because, (2Z, +, ⋅) does not have unity.
4. As the matrix ring M_n(Z) is non-commutative, it is not an integral domain.
5. The ring C[0, 1] of continuous functions on [0, 1] contains zero divisors, so it is not an integral domain. (The product of two non-zero continuous functions can be zero). For example, take f = $\begin{cases} 2x-1, & x \in [0, \frac{1}{2}] \\ 0, & x \in [\frac{1}{2}, 1] \end{cases}$ and g = $\begin{cases} 0, & x \in [0, \frac{1}{2}] \\ 1-2x, & x \in [\frac{1}{2}, 1] \end{cases}$. See that both f and g are non-zero elements of C[0, 1] but the product fg=0. In other words, f and g are zero divisors in C[0, 1].
6. The direct product ring ℤ × ℤ is not an integral domain as it contains zero divisors. Because, (1, 0) (0, 1) = (0, 0).

Main Topic: Introduction to Ring Theory

Prove that Every Finite Integral Domain is a Field

Properties of Integral Domains

An integral domain D has the following properties.

1. The set of non-zero elements of an integral domain forms a commutative semigroup under multiplication.

2. The characteristic of an integral domain is either 0 or a prime number.

3. If char D = p, a prime number, then every non-zero element of (D, +) has order p.

4. In an integral domain, the cancellation property holds.

Read Also:

Units of a Ring

Zero divisors of a ring

Idempotent and Nilpotent Elements

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