# Definition:Prime Ideal of Ring

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## 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:

### Definition 1

A **prime ideal** of $R$ is a proper ideal $P$ 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 a proper 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 $R$.

### Definition 3

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

- the complement $R \setminus P$ of $P$ in $R$ is closed under the ring product $\circ$.

## Also defined as

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

## 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

### Generalizations

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

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $22$. New Rings from Old: Exercise $22.28$