nLab
zero-divisor

Zero-divisors

Context

Algebra

Monoid theory

Zero-divisors

Idea

A zero-divisor is something that, like zero itself, can be multiplied by something nonzero to produce 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 xy=0x \cdot y = 0 or yx=0y \cdot x = 0, then y=0y = 0. An element xx is a zero-divisor if there exists y0y \ne 0 such that xy=0x \cdot y = 0 or yx=0y x = 0.

In constructive mathematics, we want \ne to be a tight apartness relation on MM in the definition of zero-divisor. We also say that xx is a strong non-zero-divisor if, whenever y0y \ne 0, then xy0x y \ne 0 and yx0y x \ne 0. (The notion of (weak) non-zero-divisor makes sense even without any apartness relation.)

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

Properties

By this definition, zero itself is a zero-divisor if and only if MM 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, 11 is not a prime number, etc. See too simple to be simple.)

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 x0x \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.

Last revised on May 21, 2021 at 19:43:07. See the history of this page for a list of all contributions to it.