= Divisorial ideal
{title2=$I_D=\mathcal O_X(-D)$}
For a <normal variety> with affine <coordinate ring> $A$ and a <Weil divisor> $D$, the fractional ideal of functions whose principal <Weil divisor> is at least $D$ is
$$
I_D=\{q\in\operatorname{Frac}(A):\operatorname{ord}_P(q)\ge\operatorname{coeff}_P D\text{ for all prime divisors }P\}\cup\{0\}.
$$
For an effective prime divisor this is its height-one <prime ideal>. If the divisor is Cartier locally, this ideal is locally principal. A nonprincipal height-one <prime ideal> in a normal <local ring> can therefore detect a non-Cartier divisor.
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