Ring theory MOC

Principal ideal domain

A principal ideal domain or PID is an integral domain in which every ideal is principal, #m/def/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 be a PID. Since 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 is a prime element.

Let be irreducible, and suppose . We have to show that either or . If we are done, so assume . Then for some . But by ^P1, is maximal among principal ideals, so . Hence there exist such that , whence

and therefore is prime.

Properties

#state/tidy | #lang/en | #SemBr

Footnotes

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