= Nonprincipal ideal generated by a prime and an indeterminate
{title2=$(p,X)\subset\mathbb Z[X]$}
For a <prime number> $p$, the <ideal> $(p,X)$ in the <polynomial ring> $\mathbb Z[X]$ is proper: the constant coefficient of every member is divisible by $p$. If $(p,X)=(g)$, then $g$ divides the nonzero constant $p$, so the additivity of <polynomial degree> over an <integral domain> makes $g$ constant. Since $g$ also divides $X$, it must divide its leading coefficient $1$; thus $g=\pm1$. This would make the ideal the whole ring, a contradiction. Therefore $\mathbb Z[X]$ is not a <principal ideal domain>, and hence not a <Euclidean domain>.
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