# Definition:Prime Ideal of Ring

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## Definition

Let $R$ be a ring.

### Definition 1

A **prime ideal** of $R$ is an ideal $P$ of $R$ such that:

- $\forall a, b \in R : a \circ b \in P \implies a \in P$ or $b \in P$

### Definition 2

A **prime ideal** of $R$ is an ideal $P$ of $R$ such that:

- $I \circ J \subseteq P \implies I \subseteq P \text { or } J \subseteq P$

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

## Also see

- Results about
**prime ideals of rings**can be found**here**.

### Special cases

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