nonUFD

NonUFD refers to a ring that is not a unique factorization domain. In an integral domain, every nonzero nonunit can be written as a product of irreducibles. If this factorization is unique up to the order of the factors and multiplication by units, the ring is a UFD (unique factorization domain). A non-UFD fails this property: some element has two essentially different factorizations into irreducibles that are not associates of one another.

A classic example is the ring Z[√-5], the ring of integers in the quadratic field Q(√-5). In this ring, the number 6 can be factored as 2·3 and also as (1+√-5)(1-√-5), with each factor irreducible and none of the factors associate to any other. These two factorizations are not equivalent, so Z[√-5] is not a UFD.

Non-UFDs occur in many rings of integers of number fields, particularly when the class group is nontrivial. In a Dedekind domain such as a ring of integers, being a UFD is equivalent to having a trivial class group. Consequently, rings with class number greater than one provide systematic sources of non-UFDs.

Beyond basic definitions, non-UFDs are studied through factorization invariants that measure how far a domain is from unique factorization, such as the variety of factorizations for a given element and related notions like elasticity and catenary degree. These ideas illuminate how factorization behaves in algebraic structures that are not UFDs.