The ring Z_{n} of integers modulo n is an integral domain when n is a prime number. In this article, we will prove that Zn is an integral domain if and only if n is a prime number.

**What is an Integral Domain?**

A non-trivial ring R is said to be an integral domain if

- R is a commutative ring with unity
- R has no divisors of zero.

A zero divisor is an element a such that ab=0 for some non-zero element b in R. Let us now prove that ℤ_{n} is an integral domain iff n is a prime number.

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## Proof

The ring ℤ_{n} is a commutative ring with unity $\overline{1}$.

**(⇒)** First suppose that ℤ_{n} is an integral domain. We need to show that n is prime. If possible assume that n is a composite number.

So n = rs for two integers r and s with 1< r, s < n.

⇒ $\overline{r} \neq \overline{0}$ and $\overline{s} \neq \overline{0}$.

Now $\overline{n}=\overline{0}$

⇒ $\overline{rs}=\overline{0}$

⇒ $\overline{r} \cdot \overline{s}=\overline{0}$.

As ℤ_{n} is an integral domain by assumption, it contains no zero divisors. This implies either $\overline{r}=\overline{0}$ or $\overline{s}=\overline{0}$. Thus we arrive at a contradiction. So n must be a prime number.

**(⇐)** Next suppose that n is a prime number.

To prove ℤ_{n} is an integral domain, we need to show that it contains no zero divisors. Let us assume that

$\overline{r} \cdot \overline{s}=\overline{0}$

⇒ $\overline{rs}=\overline{0}$.

⇒ n divides rs. As n is a prime number, it follows that either n divides r or n divides s. In other words, either $\overline{r}=\overline{0}$ or $\overline{s}=\overline{0}$. This proves that ℤ_{n} contains no zero divisors which makes it an integral domain.

Therefore, ℤ_{n} is an integral domain if and only if n is a prime number.

**Have You Read These?**

Ring Theory: Definition, Examples, Properties

Every Finite Integral Domain is a Field

## FAQs

**Q1: Is ℤ**

_{6}an integral domain?Answer: No, ℤ_{6} is not an integral domain as 6 is not a prime number. This is because ℤ_{6} contains zero divisors as $\overline{2} \cdot \overline{3}=\overline{0}$.

This article is written by Dr. T. Mandal, Ph.D in Mathematics. On Mathstoon.com you will find Maths from very basic level to advanced level. Thanks for visiting.