# Definition:Prime Spectrum of Ring

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## Definition

Let $A$ be a commutative ring with unity.

The **prime spectrum** of $A$ is the set of prime ideals $\mathfrak p$ of $A$:

- $\Spec A = \set {\mathfrak p \lhd A: \mathfrak p \text{ is prime} }$

where $\mathfrak p \lhd A$ indicates that $\mathfrak p$ is an ideal of $A$.

## Also defined as

The notation $\Spec A$ is also a shorthand for the locally ringed space:

- $\struct {\Spec A, \tau, \OO_{\Spec A} }$

where:

- $\tau$ is the Zariski topology on $\Spec A$
- $\OO_{\Spec A}$ is the structure sheaf of $\Spec A$

## Also known as

The **prime spectrum** of a commutative ring with unity is also referred to just as its **spectrum**.