# If a ring has zero divisors, is it necessarily commutative or non-commutative?

Aug 27, 2016

A ring can have zero divisors whether or not it is commutative.

#### Explanation:

Consider arithmetic modulo $4$, i.e. ${\mathbb{Z}}_{4}$ plus multiplication modulo $4$.

This is a commutative ring with $2$ being a zero divisor.

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The ring of $2 \times 2$ matrices over any ring is a non-commutative ring with zero divisors.

For example:

$\left(\begin{matrix}1 & 0 \\ 0 & 0\end{matrix}\right) \left(\begin{matrix}0 & 0 \\ 0 & 1\end{matrix}\right) = \left(\begin{matrix}0 & 0 \\ 0 & 0\end{matrix}\right)$

$\left(\begin{matrix}1 & 1 \\ 0 & 0\end{matrix}\right) \left(\begin{matrix}1 & 0 \\ 0 & 0\end{matrix}\right) = \left(\begin{matrix}1 & 0 \\ 0 & 0\end{matrix}\right) \ne \left(\begin{matrix}1 & 1 \\ 0 & 0\end{matrix}\right) = \left(\begin{matrix}1 & 0 \\ 0 & 0\end{matrix}\right) \left(\begin{matrix}1 & 1 \\ 0 & 0\end{matrix}\right)$