Let $R$ be an integral domain, $M$ a maximal ideal of $R[x]$ and $M \cap R=(0)$. Let $0\neq p\in R$. Then prove that $(M,p)=R[x]$.
2026-04-07 01:42:42.1775526162
Ideal generated by maximal ideal and some other element of integral domain.
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Since $0 \neq p \in R$ and $M \cap R = (0)$, it follows that $p$ is not in $M$. Thus, the ideal $(p, M)$ is strictly larger than $M$. By maximality of $M$, it follows that $(p, M) = R[x]$.