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 $R$.
Order $2$ Square Matrices: Example $4$
Let $R$ be the ring of square matrices of order $2$ over the real numbers.
Then:
- $\begin {bmatrix} 1 & 1 \\ 1 & 1 \end {bmatrix} \begin {bmatrix} -1 & 1 \\ 1 & -1 \end {bmatrix} = \begin {bmatrix} 0 & 0 \\ 0 & 0 \end {bmatrix}$
demonstrating that $\begin {bmatrix} 1 & 1 \\ 1 & 1 \end {bmatrix}$ and $\begin {bmatrix} -1 & 1 \\ 1 & -1 \end {bmatrix}$ are zero divisors of $R$.