Principal ideal domain

In ring the­ory, an in­te­gral do­main is a prin­ci­pal ideal do­main (or PID) if ev­ery ideal can be gen­er­ated by a sin­gle el­e­ment. That is, for ev­ery ideal \(I\) there is an el­e­ment \(i \in I\) such that \(\langle i \rangle = I\); equiv­a­lently, ev­ery el­e­ment of \(I\) is a mul­ti­ple of \(i\).

Since ideals are ker­nels of ring ho­mo­mor­phisms (proof), this is say­ing that a PID \(R\) has the spe­cial prop­erty that ev­ery ring ho­mo­mor­phism from \(R\) acts “nearly non-triv­ially”, in that the col­lec­tion of things it sends to the iden­tity is just “one par­tic­u­lar el­e­ment, and ev­ery­thing that is forced by that, but noth­ing else”.

Examples

  • Every Eu­clidean do­main is a PID. (Proof.)

  • There­fore \(\mathbb{Z}\) is a PID, be­cause it is a Eu­clidean do­main. (Its Eu­clidean func­tion is “take the mod­u­lus”.)

  • Every field is a PID be­cause ev­ery ideal is ei­ther the sin­gle­ton \(\{ 0 \}\) (i.e. gen­er­ated by \(0\)) or else is the en­tire ring (i.e. gen­er­ated by \(1\)).

  • The ring \(F[X]\) of polyno­mi­als over a field \(F\) is a PID, be­cause it is a Eu­clidean do­main. (Its Eu­clidean func­tion is “take the de­gree of the polyno­mial”.)

  • The ring of Gaus­sian in­te­gers, \(\mathbb{Z}[i]\), is a PID be­cause it is a Eu­clidean do­main. (Proof; its Eu­clidean func­tion is “take the norm”.)

  • The ring \(\mathbb{Z}[X]\) (of in­te­ger-co­effi­cient polyno­mi­als) is not a PID, be­cause the ideal \(\langle 2, X \rangle\) is not prin­ci­pal. This is an ex­am­ple of a unique fac­tori­sa­tion do­main which is not a PID. proof of this

  • The ring \(\mathbb{Z}_6\) is not a PID, be­cause it is not an in­te­gral do­main. (In­deed, \(3 \times 2 = 0\) in this ring.)

There are ex­am­ples of PIDs which are not Eu­clidean do­mains, but they are mostly un­in­ter­est­ing. One such ring is \(\mathbb{Z}[\frac{1}{2} (1+\sqrt{-19})]\). (Proof.)

Properties

  • Every PID is a unique fac­tori­sa­tion do­main. (Proof; this fact is not triv­ial.) The con­verse is false; see the case \(\mathbb{Z}[X]\) above.

  • In a PID, “prime” and “ir­re­ducible” co­in­cide. (Proof.) This fact also char­ac­ter­ises the max­i­mal ideals of PIDs.

  • Every PID is triv­ially Noethe­rian: ev­ery ideal is not just finitely gen­er­ated, but gen­er­ated by a sin­gle el­e­ment.

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