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

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

Bases: object

class ParentMethods

Bases: object

is_field(proof=True)

Return True if this ring is a field.

EXAMPLES:

Sage

sage: ZZ['x'].is_field()
False

Python

>>> from sage.all import *
>>> ZZ['x'].is_field()
False

is_integral_domain(proof=True)

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

localization(additional_units, names=None, normalize=True, category=None)

Return the localization of self at the given additional units.

EXAMPLES:

Sage

sage: R.<x, y> = GF(3)[]
sage: R.localization((x*y, x**2 + y**2))                                    # needs sage.rings.finite_rings
Multivariate Polynomial Ring in x, y over Finite Field of size 3
 localized at (y, x, x^2 + y^2)
sage: ~y in _                                                               # needs sage.rings.finite_rings
True

Python

>>> from sage.all import *
>>> R = GF(Integer(3))['x, y']; (x, y,) = R._first_ngens(2)
>>> R.localization((x*y, x**Integer(2) + y**Integer(2)))                                    # needs sage.rings.finite_rings
Multivariate Polynomial Ring in x, y over Finite Field of size 3
 localized at (y, x, x^2 + y^2)
>>> ~y in _                                                               # needs sage.rings.finite_rings
True