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zero-divisor

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 RR be a commutative ring (or any ring).

An element xx of RR is a non-zero-divisor if, whenever xy=0x y = 0 or yx=0y x = 0, then y=0y = 0. An element xx is a zero-divisor if there exists y0y \ne 0 such that xy=0x y = 0 or yx=0y x = 0.

In constructive mathematics, we want \ne to be a tight apartness relation on RR 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 RR 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 RR 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 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 xx such that x0x \ne 0).

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

Generalisations

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

Revised on June 30, 2010 05:25:32 by Toby Bartels (75.88.78.90)