Abbreviation: UFDom

A \emph{unique factorization domain} is an integral domains $D$ such that

every element is a product of irreducibles: $\forall a\in D \exists p_1,\ldots,p_r\in D, n_1,\ldots,n_r\in \mathbb{N}$ such that $a=p_1^{n_1}\cdotp_2^{n_2}\ldotsp_r^{n_r}$ and $p_i$ is irreducible for $i=1,\ldots,r$

the product is unique up to associates: $\forall \mbox{ irreducibles } p_i,q_j$ if $a=p_1^{n_1}\cdot p_2^{n_2}\ldotsp_r^{n_r}=q_1^{m_1}\cdot q_2^{m_2}\ldotsq_s^{m_s}$ then $r=s$ and each $p_i$ is an associate of some $q_j$

Example 1: $\mathbb{Z}[x]$, the ring of polynomials with integer coefficients.

$\begin{array}{lr} f(1)= &1 f(2)= &1 f(3)= &1 f(4)= &1 f(5)= &1 f(6)= &0 \end{array}$

Principal Ideal Domains

Integral domains