# Definition:Principal Open Subset of Spectrum

## Definition

Let $A$ be a commutative ring with unity.

Let $f \in A$.

The **principal open subset** determined by $f$ of the spectrum $\operatorname{Spec} A$ is the complement of the vanishing set $V \left({f}\right)$:

- $D \left({f}\right) = \operatorname{Spec} A - V \left({f}\right)$

That is, it is the set of prime ideals $\mathfrak p \subseteq A$ with $f \notin \mathfrak p$.

## Also denoted as

The **principal open subset** is also denoted $X \left({f}\right)$ or $X_f$.

## Also known as

A **principal open subset** is also known as a **basic open set**.

## Also see

## Sources

- 1969: M.F. Atiyah and I.G. MacDonald:
*Introduction to Commutative Algebra*: Chapter $1$: Rings and Ideals: Exercise 17