Prime Ideal

An Ideal is prime if and .

Proposition

Let be a proper ideal. Then is prime if and only if is an Integral domain.

Proposition

Every Maximal Ideal is prime.

Lemma

In a Principal Ideal Domain is maximal iff is irreducible. (Note, in general only one direction holds)

Lemma

In a Principal Ideal Domain, every irreducible element is prime.