Prime element

Builds on Divisibility in a commutative ring Unit of a ring

Definition

Let $R$ be a commutative ring. An element $p \in R$ is prime if it has these three properties.

$p \neq 0$ .
$p$ is not a unit.
For all $a, b \in R$ , if $p \mid a b$ , then $p \mid a$ or $p \mid b$ .

Remarks

A nonzero $p$ is prime if and only if the principal ideal $(p)$ is a prime ideal. The prime elements of $\mathbb{Z}$ are the numbers $p$ and $-p$ with $p$ a prime number. In an integral domain, prime elements are irreducible.

Reading path

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Group concept
Ring definition
Commutative ring definition
Unit of a ring definition
Divisibility in a commutative ring definition
Prime element definition

Tag	`0MNHXBSX3M`
URI	`m2://0MNHXBSX3M`
Kind	definition
Writing	W1 stated
MSC	13A05
Defines	Prime element
Uses	Divisibility in a commutative ring, Unit of a ring

Prime element

Definition

Remarks

Reading path

Where this leads

Further reading

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