Presentation of a localization at one element ($A_f\cong A[t]/(ft-1)$)

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.