= Presentation of a localization at one element
{title2=$A_f\cong A[t]/(ft-1)$}
The <localization of a ring> $A$ at the powers of $f$ is generated over $A$ by $1/f$. The presentation $A_f\cong A[t]/(ft-1)$ follows from the <universal property of localization>: both sides universally receive a <ring homomorphism> from $A$ making $f$ invertible. This proof works with <zero divisors> and includes the <zero ring> when $f$ is <nilpotent>. In particular, a <principal open subscheme> of an <affine scheme> has an inclusion which is a <morphism of finite type>.
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