zero divisor

In ordinary arithmetic, if a product of two numbers is zero then one of them is zero. Zero divisors are the elements that break this rule: nonzero things whose product is nevertheless zero. Their presence is precisely what separates a general ring from the well-behaved world of integral domains.

In a ring R, a nonzero element a is a zero divisor if there exists a nonzero element b with a·b = 0 (a left zero divisor) or b·a = 0 (a right zero divisor). An integral domain is by definition a commutative ring with no zero divisors. The existence of a zero divisor means cancellation fails: a·b = a·c no longer lets you conclude b = c.

Zero divisors are intimately tied to the structure of a commutative ring: an element is a zero divisor or zero exactly when it lies in some associated prime, and the set of zero divisors is a union of primes. Idempotents other than 0 and 1, and nonzero nilpotents, are always zero divisors. In a finite commutative ring every element is either a unit or a zero divisor, with no third option.