The 4th item answer to 4 is false. "no." If $R$ is an Artin ring, then it is Noetherian of Krull dimension zero. It follows from dimension theory that $R[X]$ is Noetherian of dimension one, i.e., not every prime ideal in $R[X]$ is maximal, so $R[X]$ can't be Artin.
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The 4th item is false. If $R$ is an Artin ring, then it is Noetherian of Krull dimension zero. It follows from dimension theory that $R[X]$ is Noetherian of dimension one, i.e., not every prime ideal in $R[X]$ is maximal, so $R[X]$ can't be Artin. |
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