Unique factorisation domain

Definition

A unique factorisation domain is an integral domain $R$ with these two properties.

1. Every nonzero element $a \in R$ that is not a unit is a product $a = p_1 p_2 \cdots p_n$ of irreducible elements, with $n \ge 1$.
2. The factorization is unique up to order and units. If $p_1 \cdots p_n = q_1 \cdots q_m$ with all $p_i, q_j$ irreducible, then $n = m$. Also, there is a permutation $\sigma$ of $\{1, \dots, n\}$ such that $p_i$ and $q_{\sigma(i)}$ are associates for each $i$.

Remarks

The integers are a unique factorisation domain: this is the fundamental theorem of arithmetic. Every principal ideal domain is a unique factorisation domain, and so is every polynomial ring over a unique factorisation domain. The ring $\mathbb{Z}[\sqrt{-5}]$ is not a unique factorisation domain. In a unique factorisation domain, irreducible elements are prime.

Explanations

L1 human

In the integers, $60 = 2 \cdot 2 \cdot 3 \cdot 5$, and every other factorization into primes uses the same primes, up to order and signs. A unique factorisation domain is a ring where this works for every nonzero nonunit. Units like $-1$ do not count as a different factorization.