Why does ruling out zero divisors matter?

Without zero divisors the cancellation law holds: if a product equals zero then one factor must be zero. This is exactly what is needed for a well-behaved theory of factorization and for embedding the ring in a field of fractions.

Are prime and irreducible elements the same thing?

Not in general. Primes are always irreducible in a domain, but irreducibles need not be prime; the failure is what makes factorization non-unique. The two coincide precisely in unique factorization domains.

Integral Domain

An integral domain is a commutative ring with identity and no zero divisors, the abstract setting in which the familiar cancellation law and notion of factorization hold.

Definition

An integral domain is a commutative ring with multiplicative identity in which the product of any two nonzero elements is nonzero, equivalently a ring with no zero divisors.

Scope

This topic covers the definition and basic properties of integral domains, the field of fractions, the hierarchy of fields, Euclidean domains, principal-ideal domains, and unique factorization domains, and the notions of irreducible and prime elements.

Core questions

What does the absence of zero divisors guarantee about cancellation and factorization?
How is an integral domain embedded in its field of fractions?
How are Euclidean, principal-ideal, and unique factorization domains related?
What is the difference between irreducible and prime elements?

Key theories

Field of fractions: Every integral domain embeds in a smallest field, its field of fractions, constructed from equivalence classes of fractions, generalizing the passage from the integers to the rationals.
Hierarchy of domains: Fields, Euclidean domains, principal-ideal domains, and unique factorization domains form a strictly descending chain of properties among integral domains, organizing how well factorization behaves.
Prime versus irreducible elements: In any integral domain prime elements are irreducible, and the two notions coincide exactly in unique factorization domains, where factorization into irreducibles is essentially unique.

Clinical relevance

Integral domains are the rings in which arithmetic behaves like that of the integers: they are the natural home of factorization theory, the rings of integers in number theory are domains, and the coordinate rings of irreducible algebraic varieties are integral domains, linking the concept to geometry.

History

The concept abstracts the arithmetic of the integers and of rings of algebraic integers studied by Dedekind and Kronecker. The systematic comparison of Euclidean, principal-ideal, and unique factorization domains emerged with the structural ring theory of the early twentieth century.

Key figures

Richard Dedekind
Leopold Kronecker
Emmy Noether

Seminal works

dummit2004
lang2002
hungerford1974

Frequently asked questions

Why does ruling out zero divisors matter?: Without zero divisors the cancellation law holds: if a product equals zero then one factor must be zero. This is exactly what is needed for a well-behaved theory of factorization and for embedding the ring in a field of fractions.
Are prime and irreducible elements the same thing?: Not in general. Primes are always irreducible in a domain, but irreducibles need not be prime; the failure is what makes factorization non-unique. The two coincide precisely in unique factorization domains.