Integral Domains — Mathematics Knowledge System

Overview

An integral domain is a commutative ring with unity 1 ≠ 0 that has no zero divisors: if a·b = 0, then a = 0 or b = 0. This single condition is what allows 'canceling' in equations (ab = ac with a ≠ 0 implies b = c) exactly as with ordinary integers, which is where the name comes from. The integers ℤ are the archetypal example; every field is automatically an integral domain, and integral domains are precisely the rings that embed into a field of fractions.

Intuition

In ℤ₆, 2×3 = 6 ≡ 0 (mod 6) even though neither 2 nor 3 is zero — those 'phantom zero products' from zero divisors are exactly the pathology that breaks cancellation and makes ℤ₆ not an integral domain. An integral domain forbids this: if a product is zero, one factor genuinely must be zero, so you can always safely divide both sides of ab=ac by a nonzero a. This is precisely the property needed to build a field of fractions the way ℚ is built from ℤ — you can't safely form a/b unless the ring has no zero divisors messing up the arithmetic.

Formal Definition

Definition

A commutative ring R with 1 ≠ 0 is an integral domain if it has no zero divisors:

a, b \in R,\ ab = 0 \implies a = 0 \text{ or } b = 0

No zero divisors

ab = ac,\ a \neq 0 \implies b = c

Cancellation property (equivalent to no zero divisors)

\text{Field} \implies \text{Integral Domain} \implies \text{Commutative ring}

Hierarchy of ring types

\operatorname{Frac}(R) = \{ a/b : a,b \in R,\ b \neq 0 \}

Field of fractions of an integral domain R

Worked Examples

Test small products: 2 × 4 = 8 ≡ 0 (mod 8), with neither 2 nor 4 equal to 0.
$2 \times 4 \equiv 0 \pmod 8$
So 2 and 4 are nonzero elements whose product is zero — they are zero divisors.
$2, 4 \text{ are zero divisors in } \mathbb{Z}_8$

Answer: ℤ₈ is NOT an integral domain, since 2·4 ≡ 0 (mod 8) with 2, 4 ≠ 0.

Practice Problems

Difficulty 4/10

Is ℤ₉ an integral domain? Justify with a specific zero-divisor pair if not.

Difficulty 5/10

Prove the cancellation law holds in any integral domain: if ab = ac and a ≠ 0, then b = c.

Difficulty 6/10

Is the ring of 2×2 matrices over ℝ an integral domain? Give a concrete zero-divisor example if not.

Quiz

Summary

An integral domain is a commutative ring with 1≠0 and no zero divisors: ab=0 implies a=0 or b=0.
This is equivalent to the cancellation law: ab=ac with a≠0 implies b=c, exactly like ordinary integer arithmetic.
ℤₙ is an integral domain iff n is prime; every field is an integral domain, and every integral domain embeds in its field of fractions.

References

WebsiteWikipedia — Integral domain