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


Special cases


Generalizations


Sources