Divisibility in a commutative ring

Definition

Let $R$ be a commutative ring and let $a, b \in R$ . The element $a$ divides $b$ , written $a \mid b$ , if there is $c \in R$ with $b = a c$ . Then $a$ is a divisor of $b$ , and $b$ is a multiple of $a$ .

Remarks

The relation $a \mid b$ holds if and only if $(b) \subseteq (a)$ for the principal ideals. Every element divides $0$ , and $0$ divides only $0$ . A unit divides every element. In $\mathbb{Z}[i]$ , $1 + i$ divides $2$ , because $2 = (1 + i)(1 - i)$ .

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Group concept
Ring definition
Commutative ring definition
Divisibility in a commutative ring definition

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Tag	`04GNK7K4VP`
URI	`m2://04GNK7K4VP`
Kind	definition
Writing	W0 stub
MSC	13A05
Defines	Divisibility in a commutative ring
Uses	Commutative ring

Divisibility in a commutative ring

Definition

Remarks

Reading path

Where this leads

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