Unique factorization domain

This article gives a basic definition in the following area: ring theory
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A unique factorization domain is an integral domain in which any non-zero element $x$ can be written as a product $x=u{p}_1\dots p_n$, where $u$ is a unit and the $p_i$s are irreducible elements uniquely, that is, if $x=u^{\prime }{q}_1\dots q_m$, with the $q_i$s irreducible, then $m=n$ and there is some permutation $\sigma \in S_n$ such that $p_i=q_{\sigma (i)}$ for all $i$.

Relation to other properties

Stronger properties

• Principal ideal domain
• Euclidean domain
• Field

Weaker properties

• Integral domain
• Ring

Examples

• $\mathbb{Z}$ is a unique factorization domain - this is the Fundamental Theorem of Arithmetic.
• The Gaussian integers $\mathbb{Z}[i]$.

Results

• The ring of integers of a number field is a unique factorization domain (also, a principal ideal domain) if and only if its ideal class group is the trivial group.