Integral domain

Symbol-free definition
A commutative unital ring is termed an integral domain (sometimes just domain) if it satisfies the following equivalent conditions:


 * It is a nonzero ring, and is cancellative
 * The zero ideal is a prime ideal
 * It is a nonzero ring, and the product of nonzero elements in nonzero
 * It is a nonzero ring, and the product of nonzero ideals is nonzero
 * It is a nonzero ring, and the set of nonzero elements is a saturated subset

Definition with symbols
A commutative unital ring $$R$$ is termed an integral domain if $$R$$ satisfies the following equivalent conditions:


 * Whenever $$ab = ac$$ and $$a$$ is not zero, $$b = c$$
 * The ideal $$0$$ is a prime ideal
 * Whenever $$ab = 0$$, either $$a = 0$$ or $$b = 0$$

Stronger properties

 * Field

Particular kinds of integral domains
Refer Category: Properties of integral domains

Weaker properties

 * Irreducible ring
 * Reduced ring

Metaproperties
The polynomial ring over an integral domain is again an integral domain.

Any subring of an integral domain is an integral domain. In fact, a commutative unital ring is an integral domain iff it occursas as a subring of a field.

A quotient of an integral domain by an ideal need not be an integral domain; to see this, note that the quotient by an ideal is an integral domain iff the ideal is a prime ideal. Thus, the quotient of $$\mathbb{Z}$$ by the non-prime ideal $$6\mathbb{Z}$$ is not an integral domain.

The localization of an integral domain at any prime ideal is still an integral domain. More generally, the localization at any multiplicatively closed subset not containing zero, continues to be an integral domain. However, the converse is not true. If a commutative unital ring has the property that its localizations at all primes are integral domains, we can only conclude that it is a direct product of integral domains, and cannot deduce that it is itself an integral domain.

Effect of property operators
Any commutative unital ring can be expressed as a quotient of an integral domain by an ideal. In fact, any commutative unital ring can be expressed as a quotient of a polynomial ring over integers (a free commutative unital ring) by a suitable ideal.