# 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 (ℤn, +, ⋅) 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 Mn(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.

Units of a Ring

Zero divisors of a ring

Idempotent and Nilpotent Elements

## FAQs

Q1: What are integral domains?

Answer: A non-trivial commutative ring with unity is called an integral domain it does not contain zero divisors. For example, (ℝ, +, ⋅), (ℚ, +, ⋅), (ℤ, +, ⋅) are integral domains.

Q2: Is Z4 an integral domain?

Answer: No, Z4 is not an integral domain as it contains zero divisors. Note that $\bar{2}$ is a zero divisor Z4, because $\bar{2} \cdot \bar{2}=\bar{0}$.

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