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 pi of R and a unit u:
- x = u p1 p2 ... pn with n≥0
and this representation is unique in the following sense: If q1,...,qm are irreducible elements of R and w is a unit such that
- x = w q1 q2 ... qm with m≥0,
then m = n and there exists a bijective map φ : {1,...,n} -> {1,...,m} such that pi is associated to qφ(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.
Read more about this topic: Unique Factorization Domain
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