nLab
zero-divisor

Idea

A zero-divisor is something that, like zero itself, can be multiplied by something nonzero to produce zero as a product.

Definitions

Let $R$ be a commutative ring (or any ring).

An element $x$ of $R$ is a non-zero-divisor if, whenever $xy=0$ or $yx=0$, then $y=0$. An element $x$ is a zero-divisor if there exists $y\ne 0$ such that $xy=0$ or $yx=0$.

In constructive mathematics, we want $\ne$ to be a tight apartness relation on $R$ in the definition of zero-divisor. We also say that $x$ is a strong non-zero-divisor if, whenever $y\ne 0$, then $xy\ne 0$ and $yx\ne 0$. (The notion of (weak) non-zero-divisor makes sense even without any apartness relation.)

If $R$ 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.

Properties

By this definition, zero itself is a zero-divisor if and only if $R$ 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, $1$ 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 $x$ such that $x\ne 0$).

The non-zero-divisors of any rig $R$ form a monoid under multiplication, which may be denoted $R^{\times }$. Note that if $R$ happens to be a field, then this $R^{\times }$ agrees with the usual notation $R^{\times }$ for the group of invertible elements of $R$, but $R^{\times }$ is not a group in general. (We may use $R^{÷}$ or $R^{*}$ for the group of invertible elements.)

Generalisations

If $I$ is any ideal of $R$, then we can generalise from a zero-divisor to an $I$-divisor. In a way, this is nothing new; $x$ is an $I$-divisor in $R$ if and only if $[x]$ is a zero-divisor in $R/I$. Ultimately, this is related to the notion of divisor? in algebraic geometry.