Zero Divisor
An element \(a\) in a ring \(R\) is called a zero divisor if there exists a non-zero element \(b\) such that
Note that this is a generalisation of the ideas of a left zero divisor (\(ab = 0\)) and a right zero divisor (\(ba = 0\)) in non-commutative rings.
Assuming the ring is not just the zero non-unital ring \(R = \{0\}\) then \(0\) is always a zero divisor. This terminology may seem confusing given that of course one cannot divide by zero, however this definition exists in the absence of a proper notion of division, which doesn't exist in general rings.
A ring which has no non-zero zero divisors is said to have the zero product property and is called a domain.
\(2 \times 2 \equiv 0 \pmod 4\) so \(2\) is a zero divisor in \(\mathbb{Z}_4\).
The ring of \(n \times n\) matrices over a field has zero divisors such as