Category:Examples of Integral Domains

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This category contains examples of Integral Domain.

Definition 1

An integral domain $\struct {D, +, \circ}$ is:

a commutative ring which is non-null
with a unity
in which there are no (proper) zero divisors, that is:
$\forall x, y \in D: x \circ y = 0_D \implies x = 0_D \text{ or } y = 0_D$

that is, in which all non-zero elements are cancellable.

Definition 2

An integral domain $\left({D, +, \circ}\right)$ is a commutative ring such that $\left({D^*, \circ}\right)$ is a monoid, all of whose elements are cancellable.

In this context, $D^*$ denotes the ring $D$ without zero: $D \setminus \left\{{0_D}\right\}$.