Finite domain $\Rightarrow$ field; field $\Longleftrightarrow R[x]$ is a PID; UFD $\Longleftrightarrow R[x]$ is a UFD; PID $\Rightarrow$ Noetherian.Suggest a slogan
Let \(R\) be a commutative ring with \(1\).
If \(R\) is a commutative division ring, then \(R\) is a field.
If \(R\) is a finite integral domain, then \(R\) is a field.
\(R\) is a field if and only if \(R[x]\) is a principal ideal domain.
\(R\) is a field if and only if \(R\) is a simple ring with \(1 \neq 0\).
\(R\) is a unique factorization domain if and only if \(R[x]\) is a unique factorization domain.
If \(R\) is a principal ideal domain, then \(R[x]\) is a unique factorization domain.
If \(R\) is a principal ideal domain, then \(R\) is Noetherian.