Module scirust::algebra::structure::integral_domain

Defines the integral domain algebraic structure.

There are a number of equivalent definitions of integral domain:

• An integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero.
• An integral domain is a nonzero commutative ring with no nonzero zero divisors.
• An integral domain is a commutative ring in which the zero ideal {0} is a prime ideal.
• An integral domain is a commutative ring for which every non-zero element is cancellable under multiplication.
• An integral domain is a ring for which the set of nonzero elements is a commutative monoid under multiplication (because the monoid is closed under multiplication).
• An integral domain is a ring that is (isomorphic to) a subring of a field. (This implies it is a nonzero commutative ring.)
• An integral domain is a nonzero commutative ring in which for every nonzero element r, the function that maps each element x of the ring to the product xr is injective. Elements r with this property are called regular, so it is equivalent to require that every nonzero element of the ring be regular.

References:

• http://en.wikipedia.org/wiki/Algebraic_structure
• http://en.wikipedia.org/wiki/Ring_(mathematics)
• http://en.wikipedia.org/wiki/Integral_domain

Traits

IntegralDomain	Marker trait for integral domains with full equivalence
IntegralDomainPartial	Marker trait for integral domains with partial equivalence