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:

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

• Definition:Prime Ideal of Commutative and Unitary Ring

• Ring with Unity has Prime Ideal
• Definition:Maximal Ideal of Ring
• Definition:Spectrum of Ring

• 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 $22.28$
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): prime ideal