zero-divisor in nLab

Zero-divisors

Zero-divisors

Idea

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

Definitions

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

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

By this definition, zero itself is a zero-divisor if and only if MM 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:

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 xx is a strong non-zero-divisor if, whenever y≠0y \ne 0, then x⋅y≠0x \cdot y \ne 0 and y⋅x≠0y \cdot x \ne 0.

If MM 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 xx such that x≠0x \ne 0).

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

Generalisations

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

• 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)