A zero-divisor is something that, like zero itself, can be multiplied by something nonzero to produce zero as a product.
An element of is a non-zero-divisor if, whenever or , then . An element is a zero-divisor if there exists such that or .
In constructive mathematics, we want to be a tight apartness relation on in the definition of zero-divisor. We also say that is a strong non-zero-divisor if, whenever , then and . (The notion of (weak) non-zero-divisor makes sense even without any apartness relation.)
If is (or may be) non-commutative, then we may distinguish left and right (non)-zero-divisors in the usual way.
Note that, in a ring, an element is a non-zero-divisor if and only if the operation of multiplication by that element is injective. This is probably the right definition of zero-divisor to use in a rig, even though then it no longer literally has anything to do with being a divisor of zero.
By this definition, zero itself is a zero-divisor if and only if is non-trivial. (Some authorities will differ on this point, but if you think about it, this is clearly the correct definition, by the same principle that the trivial ring is not a field, is not a prime number, etc. See too simple to be simple.)
An integral domain is precisely a commutative ring in which zero is the unique zero-divisor (or constructively, in which the strong non-zero-divisors are precisely the strong non-zero elements, that is those elements such that ).
The non-zero-divisors of any rig form a monoid under multiplication, which may be denoted . Note that if happens to be a field, then this agrees with the usual notation for the group of invertible elements of , but is not a group in general. (We may use or for the group of invertible elements.)
If is any ideal of , then we can generalise from a zero-divisor to an -divisor. In a way, this is nothing new; is an -divisor in if and only if is a zero-divisor in . Ultimately, this is related to the notion of divisor in algebraic geometry.