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  "body": "\n## Proof\n\nLet $R$ be a [[principal ideal domain|0RZYDVZ475]]. A factorisation of $a$ means $a = p_1 \\cdots p_n$ with $n \\ge 1$ and all $p_i$ [[irreducible|052SDEPZH7]].\n\nFirst we show that factorisations exist.\n\n1. For $b \\in R$, the set $b R = \\set{b t : t \\in R}$ is an ideal that contains $b$. Every ideal containing $b$ contains $b R$. So the [[ideal generated by|030TZ0DQWT]] $b$ is $(b) = b R$.\n2. Let $a = b c$ with $a \\neq 0$ and $c$ not a [[unit|019NFGFPR5]]. Then $(a) \\subseteq (b)$. If $(b) \\subseteq (a)$, then $b = a t$ for some $t$, so $a = a t c$.\n3. In that case [[cancellation|0GJM24EXP9]] gives $t c = 1$, and $c$ is a unit, which is false. So $(a) \\subsetneq (b)$.\n4. Let $(a_0) \\subsetneq (a_1) \\subsetneq (a_2) \\subsetneq \\cdots$ be an infinite chain of principal ideals. Let $J$ be the union of the ideals $(a_n)$.\n5. If $x \\in (a_i)$ and $y \\in (a_j)$, then both lie in $(a_k)$ with $k = \\max(i, j)$. So $x - y \\in J$ and $r x \\in J$ for $r \\in R$. Thus $J$ is an [[ideal|0MYSA6MYNA]].\n6. Since $R$ is a principal ideal domain, $J = (d)$ for some $d$. Then $d \\in (a_N)$ for some $N$, so $(a_{N+1}) \\subseteq J = (d) \\subseteq (a_N)$.\n7. This contradicts $(a_N) \\subsetneq (a_{N+1})$. So $R$ has no infinite strictly increasing chain of principal ideals.\n8. Call an element bad if it is nonzero, not a unit, and has no factorisation. Suppose that some $a_0 \\in R$ is bad.\n9. The element $a_0$ is not irreducible, because an irreducible element is a factorisation of length $1$. So $a_0 = b c$ with $b$ and $c$ not units. Both are nonzero, since $a_0 \\neq 0$.\n10. If $b$ and $c$ both had factorisations, then joining them would give a factorisation of $a_0$. So $b$ or $c$ is bad. Call it $a_1$, and call the other factor $e$.\n11. Then $a_0 = a_1 e$ with $e$ not a unit. By steps 2 and 3, $(a_0) \\subsetneq (a_1)$.\n12. Repeating steps 9 to 11 with $a_1$, then $a_2$, and so on, gives an infinite chain $(a_0) \\subsetneq (a_1) \\subsetneq \\cdots$. This contradicts step 7.\n13. So no element is bad. Every nonzero element that is not a unit has a factorisation.\n\nNow we show uniqueness.\n\n14. Let $p$ be irreducible. By [[the theorem on irreducible elements in a PID|095ZHQ7WCJ]], $(p)$ is a maximal ideal. Since [[maximal ideals are prime|05WAF5T0GZ]], $(p)$ is a [[prime ideal|06HC6HK0Z8]].\n15. So if $p \\mid a b$, then $a b \\in (p)$, hence $a \\in (p)$ or $b \\in (p)$. Since $p \\neq 0$ and $p$ is not a unit, $p$ is a [[prime element|0MNHXBSX3M]].\n16. If a prime $p$ divides $q_1 \\cdots q_m$, then $p$ divides some $q_j$. This follows by induction on $m$, writing $q_1 \\cdots q_m = (q_1 \\cdots q_{m-1}) q_m$.\n17. If $q$ is irreducible and $u$ is a unit, then $u q$ is irreducible. It is nonzero and not a unit, since $q = u^{-1} (u q)$. If $u q = x y$, then $q = x (u^{-1} y)$, so $x$ or $y$ is a unit.\n18. [[Associates|053E86NWVK]] form an equivalence relation: if $b = u a$ and $c = v b$ with units $u, v$, then $c = (v u) a$ and $a = u^{-1} b$.\n19. Let $p_1 \\cdots p_n = q_1 \\cdots q_m$ with all $p_i, q_k$ irreducible and $n, m \\ge 1$. By induction on $n$, we prove that $n = m$ and that some permutation $\\sigma$ makes $p_i$ and $q_{\\sigma(i)}$ associates.\n20. By steps 15 and 16, $p_1$ divides some $q_j$, say $q_j = p_1 u$. Since $q_j$ is irreducible and $p_1$ is not a unit, $u$ is a unit. So $p_1$ and $q_j$ are associates.\n21. Cancelling $p_1 \\neq 0$ gives $p_2 \\cdots p_n = u \\prod_{k \\neq j} q_k$. Here an empty product is $1$.\n22. Let $n = 1$. If $m \\ge 2$, step 21 gives $1 = u \\prod_{k \\neq j} q_k$, so each $q_k$ with $k \\neq j$ is a unit. This is false, so $m = 1$, and $\\sigma(1) = 1$ works.\n23. Let $n \\ge 2$. If $m = 1$, step 21 gives $p_2 (p_3 \\cdots p_n) = u$, so $p_2$ is a unit. This is false, so $m \\ge 2$.\n24. Choose $k_0 \\neq j$ and replace $q_{k_0}$ by $u q_{k_0}$, which is irreducible by step 17. Then step 21 writes $p_2 \\cdots p_n$ as a product of $m - 1$ irreducibles, each an associate of one $q_k$ with $k \\neq j$.\n25. By the induction hypothesis, $n - 1 = m - 1$, and there is a bijection $\\tau$ from $\\set{2, \\dots, n}$ to $\\set{k : k \\neq j}$ with $p_i$ an associate of $q_{\\tau(i)}$, using step 18.\n26. Put $\\sigma(1) = j$ and $\\sigma(i) = \\tau(i)$ for $i \\ge 2$. Then $n = m$, $\\sigma$ is a permutation, and $p_i$ and $q_{\\sigma(i)}$ are associates for each $i$.\n27. By steps 13 and 26, $R$ is a [[unique factorisation domain|06YMXJMXYS]].\n\n## Idea\n\nExistence: if an element had no factorisation, we could keep splitting off a bad factor forever. This would give an infinite strictly increasing chain of principal ideals, and the union of such a chain is principal, which stops the chain. Uniqueness: in a principal ideal domain irreducible elements are prime. So each irreducible factor on one side divides, and so matches, a factor on the other side. We cancel it and use induction.\n",
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