# Definition(s) > [!NOTE] Definition (Unique Factorisation Domain) \[MA268\] > Let $R$ be an [[Integral Domain|integral domain]]. We say that $R$ is a unique factorisation domain (or UFD) iff > > (1) for all $a\in R\setminus\{ 0 \}$, there exists [[Irreducible Elements of Integral Domain|irreducible]] $p_{i}$ and a [[Unit in a Ring|unit]] $u$ such that $a=up_{1}p_{2}\cdots p_{r}$; > > (2) whenever $up_{1}p_{2}\cdots p_{r}=vq_{1}q_{2}\cdots q_{s}$ where $p_{i},q_{j}$ are irreducible and $u,v$ are units, then $r=s$ and we can reorder $q_{j}$ so that for $i=1,2,\dots,r$, $p_{i}$ and $q_{i}$ are [[Associates in Integral Domain are Necessarily Unit Multiples of Each Other|associates]]. # Properties By [[Primes and Irreducibles are The Same in Unique Factorisation Domains]], ... # Reference(s)