Example of zero divisors

 In a commutative ring

$R$ with identity, a zero divisor is a nonzero element $a \in R$ such that there exists a nonzero element $b \in R$ with:

$$a \cdot b = 0$$

Example 1: Integers Modulo 6

Consider the ring $\mathbb{Z}/6\mathbb{Z} = \{0,1,2,3,4,5\}$ under multiplication mod 6.

• $2 \times 3 = 6 \equiv 0 \mod 6$

Thus, $2$ and $3$ are zero divisors in $\mathbb{Z}/6\mathbb{Z}$.

Example 2: $2 \times 2$ Matrices

In the ring of $2 \times 2$ matrices over $\mathbb{R}$, consider:

$$A = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix}$$

Then:

$$A B = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$$

Since neither $A$ nor $B$ is the zero matrix, both are zero divisors.