Zero divisor

Let $R$ be a ring. A nonzero element $a\in R$ is a left zero divisor if there exists $b\ne 0$ with $ab=0$, and a right zero divisor if there exists $c\ne 0$ with $ca=0$. In a commutative ring , these notions coincide and one simply says zero divisor.

A ring has no nonzero zero divisors precisely when it is an integral domain (in the commutative, unital convention). Zero divisors are detected by nontrivial annihilators and obstruct cancellation.

Examples:

• In $\mathbb Z/6\mathbb Z$, the class of $2$ is a zero divisor since $2\cdot 3=0$.
• In $k[x,y]/(xy)$, the classes of $x$ and $y$ are nonzero zero divisors.
• In $\mathbb Z$, no nonzero element is a zero divisor.