A zero divisor of a ring (R, +, ⋅) is an element such that its product with a non-zero element is equal to 0. In this article, we will learn about the divisors of zero in a ring with examples, and how to find zero divisors.

Table of Contents

## Definition of a Zero Divisor

Let (R, +, ⋅) be a ring. A non-zero element a ∈ R is called a zero divisor if there exists a non-zero element b in R such that a⋅b = 0, or there exists a non-zero element c in R such that c⋅a = 0.

When a⋅b = 0, a is called a left divisor of zero whereas c⋅a = 0 implies that a is a right divisor of zero.

Example: As the product of two non-zero integers is always non-zero, so the ring (ℤ, +, ⋅) does not have any divisor of zero.

## Examples

- The ring (ℝ, +, ⋅) contains no zero divisors.
- As $\bar{2} \cdot \bar{3}=\bar{0}$ in the ring Z
_{6}, the elements $\bar{2}$ and $\bar{3}$ are zero divisors in Z_{6}. - (Z
_{p}, +, ⋅) does not have zero divisors when p is a prime number.

## Properties

The zero divisors in a ring have the following properties.

- If R is a commutative ring, then every left zero divisor is also a right zero divisor.
- For a prime p, the ring (Z
_{p}, +, ⋅) of integers modulo p does not contain any zero divisors. - A ring R has no zero divisors if and only if the cancellation law holds in R.
- Let R be a non-trivial ring such that it has no zero divisors. Then R must be a ring with unity.
- If R is a finite ring with no zero divisors, then every non-zero element of R is a unit.

## How to Find Zero Divisors

In the ring (Z_{n}, +, ⋅), an element $\bar{m}$ is a zero divisor if and only if gcd(m, n) ≠ 1.Using this fact, the divisors of zero in (Z_{n}, +, ⋅) are given by {$\bar{m}$ **∈** Z_{n} : m is a divisor of n}.

Hence, the number of zero divisors in (Z_{n}, +, ⋅) is equal to φ(n) where φ denotes the Euler-Phi function.

**Example:** the set of zero divisors of (Z_{10}, +, ⋅) = {$\bar{1}, \bar{2}, \bar{5}, \bar{10}$}

**Related Topics:** Introduction to Ring Theory

## Solved Problems

**Question 1:** Prove that the ring of matrices R = $\Big\{ \begin{pmatrix} a & b \\ 2b & a \end{pmatrix} : a, b \in \mathbb{R}\Big\}$ contains zero divisors.

**Solution:**

Let A = $\begin{pmatrix} a & b \\ 2b & a \end{pmatrix}$ be a non-zero element in R. Thus either a or b is non-zero.

To find zero divisors, let AB = O, the zero matrix, for some B = $\begin{pmatrix} x & y \\ 2y & x \end{pmatrix}$

The AB = O implies that

ax + 2by = 0

bx + ay = 0.

This is a homogeneous system of equations. Thus, for non-zero solutions, we must have that $\begin{vmatrix} a & 2b \\ b & a \end{vmatrix}=0$.

⇒ a^{2} – 2b^{2} = 0.

Take a=√2, and b=1.

So there exists a non-zero B in R such that AB=O. Hence, the ring R contains zero divisors.

## FAQs

**Q1: What are divisors of zero in a ring?**

Answer: A zero divisor a of a ring R is a non-zero element such that a⋅b = 0 for some non-zero b in R. For example, $\bar{2}$ is a zero divisor in Z_{6} as $\bar{2} \cdot \bar{3}=\bar{0}$.