Unique Factorization Domain - Definition

Definition

Formally, a unique factorization domain is defined to be an integral domain R in which every non-zero x of R can be written as a product (an empty product for the unit) of irreducible elements p_i of R and a unit u:

x = u p₁ p₂ ... p_n with n≥0

and this representation is unique in the following sense: If q₁,...,q_m are irreducible elements of R and w is a unit such that

x = w q₁ q₂ ... q_m with m≥0,

then m = n and there exists a bijective map φ : {1,...,n} -> {1,...,m} such that p_i is associated to q_φ(i) for i ∈ {1, ..., n}.

The uniqueness part is usually hard to verify, which is why the following equivalent definition is useful:

A unique factorization domain is an integral domain R in which every non-zero element can be written as a product of a unit and prime elements of R.