Unique factorization domain (awaiting review)

A unique factorization domein, or UFD, is an integral domain $R$ such that for every non-zero non-unit $a\in R$ there exists irreducibles $p_1, \ldots, p_n$ such that \[ a = p_1\cdots p_n,\] and this representation is unique up to the order of the factors and up to associates.