Unique factorization domain

An integral domain \(R\) is a unique factorization domain if every nonzero \(x\in R\) can be written as \[ x=u\prod_{i=1}^n p_i \] with \(n\geq 0\), \(u\in R^\times\) a unit, and \(p_1,\ldots,p_n\) irreducible, and this decomposition is unique up to units and order: if also \(x=v\prod_{j=1}^m q_j\) with \(v\in R^\times\) and each \(q_j\) irreducible, then \(m=n\) and there is a permutation \(\sigma\) of \(\theset{1,\ldots,n}\) such that \(p_i\) and \(q_{\sigma(i)}\) are associates for every \(i\).