Category:Prime Ideals of Commutative and Unitary Rings

This category contains results about Prime Ideals of Commutative and Unitary Rings.
Definitions specific to this category can be found in Definitions/Prime Ideals of Commutative and Unitary Rings.

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 $\circ$.