Zero Divisor of Ring/Examples

Examples of Zero Divisors of Rings

Order $2$ Square Matrices: Example $1$

Let $R$ be the ring square matrices of order $2$ over a field with unity $1$ and zero $0$.

Let:

\(\ds \mathbf A\)

\(=\)

\(\ds \begin {bmatrix} 1 & 0 \\ 0 & 0 \end {bmatrix}\)

\(\ds \mathbf B\)

\(=\)

\(\ds \begin {bmatrix} 0 & 0 \\ 0 & 1 \end {bmatrix}\)

Then:

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

Thus both $\mathbf A$ and $\mathbf B$ are zero divisors of $R$.

Order $2$ Square Matrices: Example $2$

Let $R$ be the ring square matrices of order $2$ over the real numbers.

Then:

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

demonstrating that $\begin {bmatrix} 0 & 1 \\ 0 & 0 \end {bmatrix}$ and $\begin {bmatrix} 1 & 0 \\ 0 & 0 \end {bmatrix}$ are zero divisors of $R$.

Order $2$ Square Matrices: Example $3$

Let $R$ be the ring square matrices of order $2$ over the real numbers.

Then:

$\begin {bmatrix} 0 & 0 \\ 1 & 1 \end {bmatrix} \begin {bmatrix} 0 & 1 \\ 0 & -1 \end {bmatrix} = \begin {bmatrix} 0 & 0 \\ 0 & 0 \end {bmatrix}$

demonstrating that $\begin {bmatrix} 0 & 0 \\ 1 & 1 \end {bmatrix}$ and $\begin {bmatrix} 0 & 1 \\ 0 & -1 \end {bmatrix}$ are zero divisors of $R$.