Unique factorization domains appear in the following chain of class inclusions:

Definition

Formally, a unique factorization domain is defined to be an integral domain R in which every non-zero element x of R which is not a unit can be written as a finite product of irreducible elements p_i of R:
: x = p₁ p₂ ··· p_n with ''n'' ≥ 1and this

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