Unique factorization domain

An integral domain in which every nonzero nonunit factors into irreducible elements uniquely up to units and order. In such a domain, every irreducible element is prime.

Integral domain: A commutative ring with identity in which the product of two nonzero elements is nonzero. Unique factorization domains are integral domains with an additional factorization property.
Euclid's lemma: The statement that a prime dividing a product divides at least one of its factors. It lets prime factors be matched across two factorizations.
Integers: The ring of whole numbers under addition and multiplication. The integers are the standard example, with ordinary prime factorization.
Bézout domain: An integral domain in which every finitely generated ideal is principal. Bézout domains need not have unique factorization, despite their strong ideal property.
Unit (ring theory): An element of a ring that has a multiplicative inverse. Units are ignored when comparing factorizations in this domain.
Noetherian ring: A ring in which every ascending chain of ideals eventually stabilizes. Noetherian conditions can ensure that factorizations into irreducibles exist.
Polynomial ring: A ring whose elements are polynomials with coefficients in a specified ring. Polynomials over a field form a unique factorization domain.
GCD domain: An integral domain in which every pair of nonzero elements has a greatest common divisor. A gcd domain need not have factorizations into irreducibles.
Associates (ring theory): Two ring elements are associates when one is the product of the other and a unit. Associate factors count as the same factor up to multiplication by a unit.
Ascending chain condition on principal ideals: The condition that every ascending chain of principal ideals eventually stabilizes. This condition guarantees irreducible factorizations in an integral domain.

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