On principal non-maximal ideal

Question

Let $R$ be an integral domain but $R$ is not a field. Prove that, in $R[x]$, $\langle x\rangle$ is maximal principal ideal (that is, maximal among principal ideals) but not a maximal ideal.

Answer 1

Since $R$ is not a field, there is nonunit element $a\in R$. Consider the ideal $$ \langle a,x\rangle =\{ra+sx:r,s\in R[x]\}. $$

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Add. To prove $(x)\subsetneq(a,x)$, check $a\notin (x)$.