= Integer polynomial ring is not a principal ideal domain
{title2=$\mathbb Z[X]$}
The <polynomial ring> $\mathbb Z[X]$ is a <unique factorization domain> by <Gauss lemma for polynomials>, but its <ideal> $(2,X)$ is not principal. Any generator would divide both 2 and $X$, hence be a unit: a divisor of 2 has degree zero, and a constant dividing $X$ is $\pm1$. Yet evaluation at zero followed by reduction modulo 2 annihilates the ideal and not 1. It embeds in the <principal ideal domain> $\mathbb Q[X]$, illustrating that the property need not pass to a subring.
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