Noetherian domain implies every prime ideal is generated by finitely many irreducible elements

Statement
In a fact about::Noetherian domain (i.e., a fact about::Noetherian ring that is also an fact about::integral domain) every fact about::prime ideal is generated by finitely many fact about::irreducible elements.

Related facts

 * Unique factorization and Noetherian implies every prime ideal is generated by finitely many prime elements
 * Unique factorization implies every nonzero prime ideal contains a prime element
 * ACCP implies every nonzero prime ideal contains an irreducible element

Facts used

 * 1) uses::Noetherian implies ACCP
 * 2) uses::ACCP implies every nonzero prime ideal contains an irreducible element
 * 3) uses::ACCP implies every element has a factorization into irreducibles

Proof
Given: A unique factorization domain $$R$$. A prime ideal $$P$$ of $$R$$.

To prove: $$P = (p_1, p_2, \dots, p_n)$$ for irreducibles $$p_i$$ and some nonnegative integer $$n$$.

Proof:


 * 1) We do the construction inductively. Suppose we have a collection of pairwise distinct primes $$p_1, p_2, \dots, p_i$$ in $$P$$ ($$i$$ could be zero, which is covered in the general case, but can also be proved separately as shown in facts (1), (2)). We show that if $$(p_1, p_2, \dots, p_i) \ne P$$, there exists an irreducible $$p_{i+1} \in P \setminus \{ p_1, p_2, \dots, p_i \}$$:
 * 2) For this, pick $$a \in P \setminus (p_1, p_2, \dots, p_i)$$. $$a$$ has an irreducible factorization in $$R$$ (this follows from facts (1), (3)).
 * 3) Since $$P$$ is prime, at least one of the irreducible factors of $$a$$ is in $$P$$. Call this irreducible factor $$p_{i + 1}$$.
 * 4) If $$p_{i + 1} \in (p_1, p_2, \dots, p_i)$$, then $$a \in (p_1, p_2, \dots, p_i)$$ because $$a$$ is a multiple of $$p_{i+1}$$. This contradicts the way we picked $$a$$. Thus, $$p_{i+1} \in P \setminus (p_1, p_2, \dots, p_i)$$, and we have the required induction step.
 * 5) We thus have, for the prime ideal $$P$$, a strictly increasing chain of ideals $$(p_1) \subset (p_1,p_2) \subset \dots $$ with $$p_i$$ irreducible, that terminates at $$p_n$$ if and only if $$(p_1,p_2, \dots, p_n) = P$$. Since $$R$$ is Noetherian, the sequence must terminate at some finite stage, forcing $$P = (p_1, p_2, \dots, p_n)$$ for some nonnegative integer $$n$$.