Definition:Prime Ideal of Ring

Commutative Ring
Let $\struct {R, +, \circ}$ be a commutative ring with unity.

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

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

 * Equivalence of Definitions of Prime Ideal of Ring


 * Ring with Unity has Prime Ideal
 * Definition:Maximal Ideal of Ring
 * 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