Zero Divisor of Ring/Examples/Order 2 Square Matrices/Example 1

Examples of Zero Divisors of Rings

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 $R$.

Sources

• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): zero divisors
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): zero-divisor