Integral domains#

class sage.categories.integral_domains.IntegralDomains(base_category)[source]#

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

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

Python

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

class ElementMethods[source]#: Bases: object

class ParentMethods[source]#

Bases: object

is_integral_domain(proof=True)[source]#

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

EXAMPLES:

Sage

sage: ZZ.is_integral_domain()
True
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

sage: ZZ['x'].is_integral_domain()
True

Python

>>> from sage.all import *
>>> ZZ.is_integral_domain()
True
>>> QQ.is_integral_domain()
True
>>> Parent(QQ, category=IntegralDomains()).is_integral_domain()
True

>>> L = LazyLaurentSeriesRing(QQ, names=('z',)); (z,) = L._first_ngens(1)# needs sage.combinat
>>> L.is_integral_domain()                                            # needs sage.combinat
True
>>> L.is_integral_domain(proof=True)                                  # needs sage.combinat
True

>>> ZZ['x'].is_integral_domain()
True