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Is $\mathbb{Z}_6[x]$ is principal ideal domain ?

no,

we know that polynomial ring $\mathbb{F}[x]$ is field iff $\mathbb{F}$ is field

here $\mathbb{Z}_6$ is not field.

consider ideal $\langle{x,2}\rangle$ which is not principal ideal in $\mathbb{Z}_6[x]$ hence it is not PID.

please correct me if i am wrong.

dipali mali
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  • You are absolutely right. Depending on context, you might be requred to explicitly prove that $\langle x, 2\rangle$ is not principal, though. – Arthur Jul 17 '17 at 10:35
  • $\mathbb{F}[x]$ is never a field. – lhf Jul 17 '17 at 11:29

1 Answers1

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$\mathbb{Z}_6[x]$ is not a principal ideal domain because it is not even a domain.

lhf
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