Definition 10.17.1. Let $R$ be a ring.

1. The spectrum of $R$ is the set of prime ideals of $R$. It is usually denoted $\mathop{\mathrm{Spec}}(R)$.

2. Given a subset $T \subset R$ we let $V(T) \subset \mathop{\mathrm{Spec}}(R)$ be the set of primes containing $T$, i.e., $V(T) = \{ \mathfrak p \in \mathop{\mathrm{Spec}}(R) \mid \forall f\in T, f\in \mathfrak p\}$.

3. Given an element $f \in R$ we let $D(f) \subset \mathop{\mathrm{Spec}}(R)$ be the set of primes not containing $f$.

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