Question

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

Answer 1

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

Explanation:

Consider arithmetic modulo `4`, i.e. `ZZ_4` plus multiplication modulo `4`.

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

`color(white)()`
The ring of `2xx2` matrices over any ring is a non-commutative ring with zero divisors.

For example:

`((1,0),(0,0))((0,0),(0,1)) = ((0,0),(0,0))`

`((1,1),(0,0))((1,0),(0,0)) = ((1,0),(0,0)) != ((1,1),(0,0)) = ((1,0),(0,0))((1,1),(0,0))`