Proper Zero Divisor/Examples

Examples of Proper Zero Divisors

Proper Zero Divisors of Integer Multiplication Modulo $6$

Consider the multiplicative monoid of integers modulo $6$, defined by its Cayley table:

$\quad \begin{array} {r|rrrrrr} \struct {\Z_6, \times_6} & \eqclass 0 6 & \eqclass 1 6 & \eqclass 2 6 & \eqclass 3 6 & \eqclass 4 6 & \eqclass 5 6 \\ \hline \eqclass 0 6 & \eqclass 0 6 & \eqclass 0 6 & \eqclass 0 6 & \eqclass 0 6 & \eqclass 0 6 & \eqclass 0 6 \\ \eqclass 1 6 & \eqclass 0 6 & \eqclass 1 6 & \eqclass 2 6 & \eqclass 3 6 & \eqclass 4 6 & \eqclass 5 6 \\ \eqclass 2 6 & \eqclass 0 6 & \eqclass 2 6 & \eqclass 4 6 & \eqclass 0 6 & \eqclass 2 6 & \eqclass 4 6 \\ \eqclass 3 6 & \eqclass 0 6 & \eqclass 3 6 & \eqclass 0 6 & \eqclass 3 6 & \eqclass 0 6 & \eqclass 3 6 \\ \eqclass 4 6 & \eqclass 0 6 & \eqclass 4 6 & \eqclass 2 6 & \eqclass 0 6 & \eqclass 4 6 & \eqclass 2 6 \\ \eqclass 5 6 & \eqclass 0 6 & \eqclass 5 6 & \eqclass 4 6 & \eqclass 3 6 & \eqclass 2 6 & \eqclass 1 6 \end{array}$

Thus we have:

$\eqclass 2 6 \times \eqclass 3 6 = \eqclass 0 6$

and:

$\eqclass 4 6 \times \eqclass 3 6 = \eqclass 0 6$

Hence in the ring of integers modulo $6$, there are seen to be $3$ proper zero divisors: $\eqclass 2 6$, $\eqclass 3 6$ and $\eqclass 4 6$.

Proper Zero Divisors of Ring of Order 2 Complex Matrices

Consider the ring of square matrices $\struct {\map {\MM_\C} 2, +, \times}$ of order $2$ over the complex numbers $\C$.

We have:

$\begin {pmatrix} 1 & i \\ i & -1 \end {pmatrix} \begin {pmatrix} 1 & i \\ i & -1 \end {pmatrix} = \begin {pmatrix} 0 & 0 \\ 0 & 0 \end {pmatrix}$

demonstrating that $\begin {pmatrix} 1 & i \\ i & -1 \end {pmatrix}$ is a proper zero divisor of $\struct {\map {\MM_\C} 2, +, \times}$.