Unique Factorization Domains are integral domains in which nonzero elements that are not units can be written as a finite product of irreducible elements and this decomposition is unique up to associates and the order of factors. We have the following inclusions: fields $\subset$ Euclidean Domains $\subset$ Principal Ideal Domains $\subset$ Unique Factorization Domains $\subset$ integral domains with all containments being proper. Thus: fields, Euclidean Domains and Principal Ideal Domains are examples of Unique Factorization Domains. The polynomial ring $K[x]$ is a Unique Factorization Domain if and only if $K$ is a Unique Factorization Domain.
|
