Definition:Prime Ideal of Ring/Commutative and Unitary Ring

Definition

Let $\struct {R, +, \circ}$ be a commutative and unitary ring.

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 see

• Equivalence of Definitions of 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 commutative and unitary rings can be found here.