Ring theory MOC

Principal ideal domain

A principal ideal domain or PID 𝑅 is an integral domain in which every ideal is principal, ring i.e. it is also a Principal ideal ring. Every principal ideal domain is Noetherian (by ^N1) and a Unique factorization domain¹

Proof

Let 𝑅 be a PID. Since 𝑅 is Noetherian, the ^N2 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

𝑐=𝑟𝑎𝑐+𝑠𝑏𝑐∈⟨𝑎⟩

and therefore 𝑎 is prime.

Properties

• Maximal ideal iff prime ideal in a PID

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Footnotes

1. 2009. Algebra: Chapter 0, §V.2.3, pp. 254–255 ↩