Ring of integers

Statement

The integers $\mathbb{Z}$ , with their addition $+$ and multiplication $\cdot$ , form a commutative ring with identity $1$ . That is, for all integers $a$ , $b$ and $c$ :

$(a + b) + c = a + (b + c)$ and $a + b = b + a$ ;
$a + 0 = a$ , and there is an integer $-a$ with $a + (-a) = 0$ ;
$(a b) c = a (b c)$ and $a b = b a$ ;
$a \cdot 1 = a$ ;
$a (b + c) = a b + a c$ .

Remarks

The ring $\mathbb{Z}$ is not a field: for example, $2$ has no multiplicative inverse in $\mathbb{Z}$ . The ring $\mathbb{Z}$ has no zero divisors: if $a b = 0$ , then $a = 0$ or $b = 0$ . Thus it is an integral domain. Here "ring of integers" means the ring $\mathbb{Z}$ . In algebraic number theory, the same words have a more general meaning.

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Integers object
Ring of integers statement

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Tag	`0BXA4Q0JHW`
URI	`m2://0BXA4Q0JHW`
Kind	statement, theorem
Writing	W0 stub
MSC	13A99
Status	This copy of the site has no build data. Run `m2 status 0BXA4Q0JHW`.
Uses	Integers

Ring of integers

Statement

Remarks

Reading path

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