nLab zero-divisor

Idea

A zero-divisor is something that, like zero itself, when multiplied by something possibly nonzero still produces zero as a product.

Definitions

Let $M$ be a absorption monoid (such as a commutative ring or any ring).

An element $x$ of $M$ is a non-zero-divisor if, whenever $x \cdot y = 0$ or $y \cdot x = 0$, then $y = 0$.

Definition

An element $x$ is a zero-divisor if there exists $y \ne 0$ such that $x \cdot y = 0$ or $y \cdot x = 0$.

$$\mathrm{isZeroDivisor}(x) \coloneqq \exists y \in M.\neg(y = 0) \Rightarrow ((x \cdot y = 0) \vee (y \cdot x = 0))$$

By this definition, zero itself is a zero-divisor if and only if $M$ is non-trivial (see too simple to be simple)

Alternatively, one can define a zero-divisor using a weakened version of negation from Lombardi & Quitté 2010 in the definition of zero divisor:

Definition

An element $x$ is a zero-divisor if there exists an element $y$ such that if $y = 0$ then $1 = 0$, and $x \cdot y = 0$ or $y \cdot x = 0$.

$$\mathrm{isZeroDivisor}\prime(x) \coloneqq \exists y \in M.((y = 0) \Rightarrow (1 = 0)) \wedge ((x \cdot y = 0) \vee (y \cdot x = 0))$$

By this definition, zero itself is also a zero divisor in the trivial monoid.

In constructive mathematics

By the antithesis interpretation of constructive mathematics we want $\ne$ to be an arbitrary irreflexive symmetric relation and we want the monoid operation to be strongly extensional with respect to $\ne$ as well. We also say that $x$ is a strong non-zero-divisor if, whenever $y \ne 0$, then $x \cdot y \ne 0$ and $y \cdot x \ne 0$.

If $M$ is (or may be) non-commutative, then we may distinguish left and right (non)-zero-divisors in the usual way.

Properties

An integral domain is precisely a commutative ring (whose multiplicative monoid is an absorption monoid by definition) in which zero is the unique zero-divisor of the multiplicative monoid of the commutative ring (or constructively, in which the strong non-zero-divisors are precisely the strong non-zero elements in the multiplicative monoid, that is those elements $x$ such that $x \ne 0$).

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

Generalisations

If $I$ is any ideal of $M$, 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 $M$ if and only if $[x]$ is a zero-divisor in $M/I$. Ultimately, this is related to the notion of divisor in algebraic geometry.

Related concepts

• absorption monoid

• integral domain

• cancellative element

 References

• Henri Lombardi, Claude Quitté (2010): Commutative algebra: Constructive methods (Finite projective modules) Translated by Tania K. Roblo, Springer (2015) (doi:10.1007/978-94-017-9944-7, pdf)