Ideals in a polynomial ring

Mar 17, 2025 2:54 PM

Let $R$ be a commutative ring with $1$ . Use theorems in ring theory to prove:

$⟨ x ⟩$ is a prime ideal in $R [x]$ if and only if $R$ is an integral domain.
$⟨ x ⟩$ is a maximal ideal in $R [x]$ if and only if $R$ is a field.

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Let $R$ be a commutative ring with $1$. Use theorems in ring theory to prove:
\begin{enumerate}[label=\alph*)]
	\item $\langle x\rangle$ is a prime ideal in $R[x]$ if and only if $R$ is an integral domain.
	\item $\langle x\rangle$ is a maximal ideal in $R[x]$ if and only if $R$ is a field.
\end{enumerate}