Principal Ideal Domain

An Integral Domain is a PID if every Ideal is a Principal Ideal i.e. .

Lemma

Let , where is PID. Then is maximal if and only if is irreducible.

Lemma

In PID, every irreducible element is prime.

Lemma

Let be a PID and be a nested sequence of ideals. Then there is some such that for all .

Proof

Consider the union = for some . Hence find s.t. .

Theorem

Every PID is a Unique Factorisation Domain.

Proof