Definition:Prime Ideal of Ring

Definition
Let $\left({R, +, \circ}\right)$ be a ring.

Let $J$ be a proper ideal of $R$.

Then $J$ is a prime ideal :
 * $J_1 \circ J_2 \subseteq J \implies J_1 \subseteq J \text{ or } J_2 \subseteq J$

for any ideals $J_1$ and $J_2$ of $R$.

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

Also see

 * Definition:Prime Element
 * Definition:Maximal Ideal