Unique Factorization Domain

definitionring-theory

Definition

We call $R$ a unique factorization domain if,

1. $\forall r\in R$, $r\ne 0$, unit factors as $r=x_1\cdots x_n$ for some $n\ge 1$.
2. $(\ast )$ is unique up to commuting $x_i$ or multiplying them by units.

Examples

• $\mathbb{Z}$, fields $F$, Euclidean domains, principal ideal domains (eg. $F[x]$) are all unique factorization domains.
• $\mathbb{Z}[i\sqrt{3}]$ is not a unique factorization domain.
  • Claim: $2\cdot 2=4=(1+i\sqrt{3})(1-i\sqrt{3})$ is a non-unique factorization of $4=4+0i\sqrt{3}$.

Related

• Irreducible