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Proposition 3 \( p \) is irreducible if and only if \( \langle p \rangle \) is maximal amongst all principal ideals that contain \( \langle p \rangle \).

Definition 1 An ideal \( I \) of \( R \) is prime if the quotient \( R / I \) is an integral domain. It is maximal if \( R / I \) is a field.