Prime ideal

Definition

Let $R$ be a commutative ring. An ideal $P$ of $R$ is prime if it has these two properties.

$P \neq R$ .
For all $a, b \in R$ , if $a b \in P$ , then $a \in P$ or $b \in P$ .

Remarks

In $\mathbb{Z}$ , the ideal $n\mathbb{Z}$ with $n \ge 2$ is prime if and only if $n$ is a prime number. The zero ideal of $\mathbb{Z}$ is also prime. The zero ideal of $R$ is prime if and only if $R$ is an integral domain. An ideal $P$ is prime if and only if $R/P$ is an integral domain. Every maximal ideal is prime.

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Group concept
Ring definition
Commutative ring definition
Ideal definition
Prime ideal definition

Tag	`06HC6HK0Z8`
URI	`m2://06HC6HK0Z8`
Kind	definition
Writing	W1 stated
MSC	13A15
Defines	Prime ideal
Uses	Ideal, Commutative ring

Prime ideal

Definition

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