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This question already has an answer here:

Does there exist an element $p$ in a ring $R$ such that $p$ is an irreducible element but the ideal $\langle p\rangle $ is not a maximal ideal?

I could only find that $R$ is not a PID but I could not find any counterexample to the problem.Please help.

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marked as duplicate by rschwieb abstract-algebra Jan 7 '16 at 12:07

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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Take a prime number and consider it as an element in $\mathbb Z[x]$. $(p)$ is not maximal, since $\mathbb Z[x]/(p) \cong \mathbb F_p[x]$ is not a field.

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