 Zero divisor

A nonzero element $a$ of a commutative ring is a zero divisor if some nonzero $b$ satisfies $ab=0$. In a noncommutative ring, left and right zero divisors must be distinguished. The nonzero requirement on both factors is part of the usual convention.

In $\mathbb{Z}/6\mathbb{Z}$, the classes of two and three are zero divisors because their product is zero. A commutative ring with identity, $1\ne 0$, and no zero divisors is an integral domain. Cancellation by a nonzero factor is valid in an integral domain but can fail in a ring with zero divisors: modulo six, $2\cdot 1=2\cdot 4$ although $1\ne 4$.