Definition:Prime Ideal of Ring

Commutative Ring
Let $\left({R, +, \circ}\right)$ be a commutative ring with unity.

Definition 1
A prime ideal of $R$ is a proper ideal $P$ such that:
 * $\forall a,b \in R : ab \in P \implies a \in P$ or $b\in P$

Definition 2
A prime ideal of $R$ is a proper ideal $P$ such that:
 * $I \circ J \subseteq P \implies I \subseteq P \text{ or } J \subseteq P$

for any ideals $I$ and $J$ of $R$.

The set of prime ideals is called the (prime) spectrum $\operatorname{Spec}(A)$.

General ring
Let $R$ be a ring.

A prime ideal of $R$ is a proper ideal $P$ such that:
 * $I \circ J \subseteq P \implies I \subseteq P \text{ or } J \subseteq P$

for any ideals $I$ and $J$ of $R$.

Also defined as
Some sources do not require the ideal $P$ to be proper.

Also see

 * Definition:Maximal Ideal
 * Definition:Spectrum of Ring

Special cases

 * Definition:Prime Number
 * Definition:Prime Element of Ring, as shown at Prime Element iff Generates Principal Prime Ideal

Generalizations

 * Definition:Prime Ideal (Order Theory), as shown at Prime Ideal of Ring iff Prime Ideal in Lattice of Ideals