Definition:Integral Domain

From ProofWiki
Jump to: navigation, search

This page is about the concept of integral domain in ring theory. For other uses, see Definition:Domain.

Definition

Definition 1

An integral domain $\left({D, +, \circ}\right)$ 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 (from the Cancellation Law of Ring Product of Integral Domain) 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: $D \setminus \left\{{0_D}\right\}$.


Integral Domain Axioms

An integral domain is an algebraic structure $\left({D, *, \circ}\right)$, on which are defined two binary operations $\circ$ and $*$, which satisfy the following conditions:

\((A0)\)   $:$   Closure under addition      \(\displaystyle \forall a, b \in D:\) \(\displaystyle a * b \in D \)             
\((A1)\)   $:$   Associativity of addition      \(\displaystyle \forall a, b, c \in D:\) \(\displaystyle \left({a * b}\right) * c = a * \left({b * c}\right) \)             
\((A2)\)   $:$   Commutativity of addition      \(\displaystyle \forall a, b \in D:\) \(\displaystyle a * b = b * a \)             
\((A3)\)   $:$   Identity element for addition: the zero      \(\displaystyle \exists 0_D \in D: \forall a \in D:\) \(\displaystyle a * 0_D = a = 0_D * a \)             
\((A4)\)   $:$   Inverse elements for addition: negative elements      \(\displaystyle \forall a \in D: \exists a' \in D:\) \(\displaystyle a * a' = 0_D = a' * a \)             
\((M0)\)   $:$   Closure under product      \(\displaystyle \forall a, b \in D:\) \(\displaystyle a \circ b \in D \)             
\((M1)\)   $:$   Associativity of product      \(\displaystyle \forall a, b, c \in D:\) \(\displaystyle \left({a \circ b}\right) \circ c = a \circ \left({b \circ c}\right) \)             
\((D)\)   $:$   Product is distributive over addition      \(\displaystyle \forall a, b, c \in D:\) \(\displaystyle a \circ \left({b * c}\right) = \left({a \circ b}\right) * \left({a \circ c}\right) \)             
\(\displaystyle \left({a * b}\right) \circ c = \left({a \circ c}\right) * \left({b \circ c}\right) \)             
\((C)\)   $:$   Product is commutative      \(\displaystyle \forall a, b \in D:\) \(\displaystyle a \circ b = b \circ a \)             
\((U)\)   $:$   Identity element for product: the unity      \(\displaystyle \exists 1_D \in D: \forall a \in D:\) \(\displaystyle a \circ 1_D = a = 1_D \circ a \)             
\((ZD)\)   $:$   No proper zero divisors      \(\displaystyle \forall a, b \in D:\) \(\displaystyle a \circ b = 0_D \iff a = 0 \lor b = 0 \)             

These criteria are called the integral domain axioms.


Also known as

Some authors refer to this concept as simply a domain.

However, this conflicts with the concept of domain in set theory, in the context of mappings and relations.

Therefore, it is always best to refer to an integral domain, as to avoid possible confusion.


Also defined as

Some authors do not require that an integral domain be commutative.


Also see