Definition:Prime Ideal of Ring

Definition
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 $R$.

Commutative and Unitary Ring
When $\struct {R, +, \circ}$ is a commutative and unitary ring, the definition of a prime ideal can be given in a number of equivalent forms:

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

Also see

 * Definition:Prime Ideal of Commutative and Unitary 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