Ring theory MOC

Unique factorization domain

A unique factorization domain or UFD 𝑅 is an integral domain such that every nonzero element 𝑥 ∈𝑅 has a factorization as a product of irreducible elements, unique up to units and the order of factors. num

𝑥=𝑞1⋯𝑞𝑟

Every UFD is also a GCD domain.

Equivalent characterizations

Let 𝑅 be an integral domain. The following are equivalent:

1. 𝑅 is a UFD;
2. Every irreducible element in 𝑅 is prime and 𝑅 satisfies the ^N2 on principal ideals.¹

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Footnotes

1. 2009. Algebra: Chapter 0, § V.2.2, p. 253 ↩