Maximal ideals in a PID

#ring_theory

Suppose $R$ is a PID (principal ideal domain). Prove that an ideal $I \subset R$ is maximal if and only if $I = ⟨ p ⟩$ for a prime $p \in R$ . (By definition, an element $p$ is prime if whenever $p ∣ a b$ then $p ∣ a$ or $p ∣ b$ . If you use the fact that prime implies irreducible, you have to prove it.)

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Suppose $R$ is a PID (principal ideal domain). Prove that an ideal $I\subset R$ is maximal if and only if $I=\langle p\rangle$ for a prime $p\in R$. (By definition, an element $p$ is {\bfseries prime} if whenever $p\mid ab$ then $p\mid a$ or $p\mid b$. If you use the fact that prime implies irreducible, you have to prove it.)