Let 𝑅 be a PID. Since 𝑅 is Noetherian, the ACC holds in general and thus in particular for principal ideals, so invoking ^U2 it is sufficient to show that every irreducible element in 𝑅 is a prime element.

Let 𝑎 ∈𝑅 be irreducible, and suppose 𝑏𝑐 ∈⟨𝑎⟩. We have to show that either 𝑏 ∈⟨𝑎⟩ or 𝑐 ∈⟨𝑎⟩. If 𝑏 ∈⟨𝑎⟩ we are done, so assume 𝑏 ∉⟨𝑎⟩. Then ⟨𝑎⟩ ⊊⟨𝑎,𝑏⟩ =⟨𝑑⟩ for some 𝑑 ∈𝑅. But by ^P1, ⟨𝑎⟩ is maximal among principal ideals, so ⟨𝑑⟩ =⟨1⟩. Hence there exist 𝑟,𝑠 ∈𝑅 such that 𝑟𝑎 +𝑠𝑏 =1, whence

and therefore 𝑎 is prime.