Domínio de integridade

Nota: Não confundir com Domínio (matemática).

Um domínio de integridade (ou anel de integridade)é um anel $(D,+,.)$ com as seguintes propriedades adicionais:

$\exists 1\in D\ (1\neq 0\land \forall x\in D\ (1.x=x.1=x))$ (elemento neutro)
$\forall x,y\in D\ (x.y=y.x)$ (comutatividade)
$\forall x,y\in D\ (x.y=0\rightarrow (x=0\lor y=0))$ (não existem divisores de zero)

Exemplos

O conjunto dos números inteiros $\mathbb {Z}$
Todo corpo é um domínio de integridade
Analogamente, todo domínio de integridade finito é um corpo: seja a um elemento não-nulo de um domínio de integridade finito D. Então a função $f:D\to D,f(x)=ax$ é injetiva (caso contrário, f(x) = f(y), a x = a y logo a (x - y) = 0 e D teria divisores de zero), logo sobrejetiva, portanto existe b tal que f(b) = 1.
Para qualquer corpo ou domínio de integridade D, o anel dos polinômios D[x] é um domínio de integridade.
Os anéis finitos $\mathbb {Z} _{n}$ não são domínios de integridade quando n for um número composto, porque sendo $n=a\ b,$ então em $\mathbb {Z} _{n}{\mbox{ , }}a\ b=0.$ Quando n for um número primo, $\mathbb {Z} _{n}$ é um corpo (logo, é um domínio de integridade).

Corpo de frações

Ver artigo principal: Corpo de frações

Para todo domínio de integridade D existe um corpo K, $D\subseteq K,$ tal que todo elemento de K pode ser escrito da forma a/b, sendo $a,b\in D.$

Este é o corpo de frações de D, e é único no seguinte sentido algébrico: se $K_{1}$ é outro corpo $K_{1}\supseteq D$ em que todo elemento de $K_{1}$ pode ser escrito como a/b com $a,b\in D,$ então K e $K_{1}$ são isomorfos.

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