Integral domains

class sage.categories.integral_domains.IntegralDomains(base_category)

Bases: CategoryWithAxiom_singleton

The category of integral domains

An integral domain is commutative ring with no zero divisors, or equivalently a commutative domain.

EXAMPLES:

sage: C = IntegralDomains(); C
Category of integral domains
sage: sorted(C.super_categories(), key=str)
[Category of commutative rings, Category of domains]
sage: C is Domains().Commutative()
True
sage: C is Rings().Commutative().NoZeroDivisors()
True

class ElementMethods

Bases: object

class ParentMethods

Bases: object

is_integral_domain(proof=True)

Return True, since this in an object of the category of integral domains.

EXAMPLES:

sage: QQ.is_integral_domain()
True
sage: Parent(QQ, category=IntegralDomains()).is_integral_domain()
True

sage: L.<z> = LazyLaurentSeriesRing(QQ)                                 # needs sage.combinat
sage: L.is_integral_domain()                                            # needs sage.combinat
True
sage: L.is_integral_domain(proof=True)                                  # needs sage.combinat
True