Rings

class sage.categories.rings.Rings(base_category)

Bases: sage.categories.category_with_axiom.CategoryWithAxiom_singleton

The category of rings

Associative rings with unit, not necessarily commutative

EXAMPLES:

sage: Rings()
Category of rings
sage: sorted(Rings().super_categories(), key=str)
[Category of rngs, Category of semirings]

sage: sorted(Rings().axioms())
['AdditiveAssociative', 'AdditiveCommutative', 'AdditiveInverse',
 'AdditiveUnital', 'Associative', 'Distributive', 'Unital']

sage: Rings() is (CommutativeAdditiveGroups() & Monoids()).Distributive()
True
sage: Rings() is Rngs().Unital()
True
sage: Rings() is Semirings().AdditiveInverse()
True

Todo

(see: http://trac.sagemath.org/sage_trac/wiki/CategoriesRoadMap)

  • Make Rings() into a subcategory or alias of Algebras(ZZ);
  • A parent P in the category Rings() should automatically be in the category Algebras(P).
Commutative

alias of sage.categories.commutative_rings.CommutativeRings

Division

alias of sage.categories.division_rings.DivisionRings

class ElementMethods
inverse_of_unit()

Return the inverse of this element if it is a unit.

OUTPUT:

An element in the same ring as this element.

EXAMPLES:

sage: R.<x> = ZZ[]
sage: S = R.quo(x^2 + x + 1)
sage: S(1).inverse_of_unit()
1

This method fails when the element is not a unit:

sage: 2.inverse_of_unit()
Traceback (most recent call last):
...
ArithmeticError: inverse does not exist

The inverse returned is in the same ring as this element:

sage: a = -1
sage: a.parent()
Integer Ring
sage: a.inverse_of_unit().parent()
Integer Ring

Note that this is often not the case when computing inverses in other ways:

sage: (~a).parent()
Rational Field
sage: (1/a).parent()
Rational Field
is_unit()

Return whether this element is a unit in the ring.

Note

This is a generic implementation for (non-commutative) rings which only works for the one element, its additive inverse, and the zero element. Most rings should provide a more specialized implementation.

EXAMPLES:

sage: MS = MatrixSpace(ZZ, 2)
sage: MS.one().is_unit()
True
sage: MS.zero().is_unit()
False
sage: MS([1,2,3,4]).is_unit()
False
class MorphismMethods
is_injective()

Return whether or not this morphism is injective.

EXAMPLES:

This often raises a NotImplementedError as many homomorphisms do not implement this method:

sage: R.<x> = QQ[]
sage: f = R.hom([x + 1]); f
Ring endomorphism of Univariate Polynomial Ring in x over Rational Field
  Defn: x |--> x + 1
sage: f.is_injective()
Traceback (most recent call last):
...
NotImplementedError

If the domain is a field, the homomorphism is injective:

sage: K.<x> = FunctionField(QQ)
sage: L.<y> = FunctionField(QQ)
sage: f = K.hom([y]); f
Function Field morphism:
  From: Rational function field in x over Rational Field
  To:   Rational function field in y over Rational Field
  Defn: x |--> y
sage: f.is_injective()
True

Unless the codomain is the zero ring:

sage: codomain = Integers(1)
sage: f = QQ.hom([Zmod(1)(0)], check=False)
sage: f.is_injective()
False

Homomorphism from rings of characteristic zero to rings of positive characteristic can not be injective:

sage: R.<x> = ZZ[]
sage: f = R.hom([GF(3)(1)]); f
Ring morphism:
  From: Univariate Polynomial Ring in x over Integer Ring
  To:   Finite Field of size 3
  Defn: x |--> 1
sage: f.is_injective()
False

A morphism whose domain is an order in a number field is injective if the codomain has characteristic zero:

sage: K.<x> = FunctionField(QQ)
sage: f = ZZ.hom(K); f
Composite map:
  From: Integer Ring
  To:   Rational function field in x over Rational Field
  Defn:   Conversion via FractionFieldElement_1poly_field map:
          From: Integer Ring
          To:   Fraction Field of Univariate Polynomial Ring in x over Rational Field
        then
          Isomorphism:
          From: Fraction Field of Univariate Polynomial Ring in x over Rational Field
          To:   Rational function field in x over Rational Field
sage: f.is_injective()
True

A coercion to the fraction field is injective:

sage: R = ZpFM(3)
sage: R.fraction_field().coerce_map_from(R).is_injective()
True
NoZeroDivisors

alias of sage.categories.domains.Domains

class ParentMethods
bracket(x, y)

Returns the Lie bracket \([x, y] = x y - y x\) of \(x\) and \(y\).

INPUT:

  • x, y – elements of self

EXAMPLES:

sage: F = AlgebrasWithBasis(QQ).example()
sage: F
An example of an algebra with basis: the free algebra on the generators ('a', 'b', 'c') over Rational Field
sage: a,b,c = F.algebra_generators()
sage: F.bracket(a,b)
B[word: ab] - B[word: ba]

This measures the default of commutation between \(x\) and \(y\). \(F\) endowed with the bracket operation is a Lie algebra; in particular, it satisfies Jacobi’s identity:

sage: F.bracket( F.bracket(a,b), c) + F.bracket(F.bracket(b,c),a) + F.bracket(F.bracket(c,a),b)
0
characteristic()

Return the characteristic of this ring.

EXAMPLES:

sage: QQ.characteristic()
0
sage: GF(19).characteristic()
19
sage: Integers(8).characteristic()
8
sage: Zp(5).characteristic()
0
ideal(*args, **kwds)

Create an ideal of this ring.

NOTE:

The code is copied from the base class Ring. This is because there are rings that do not inherit from that class, such as matrix algebras. See trac ticket #7797.

INPUT:

  • An element or a list/tuple/sequence of elements.
  • coerce (optional bool, default True): First coerce the elements into this ring.
  • side, optional string, one of "twosided" (default), "left", "right": determines whether the resulting ideal is twosided, a left ideal or a right ideal.

EXAMPLES:

sage: MS = MatrixSpace(QQ,2,2)
sage: isinstance(MS,Ring)
False
sage: MS in Rings()
True
sage: MS.ideal(2)
Twosided Ideal
(
  [2 0]
  [0 2]
)
 of Full MatrixSpace of 2 by 2 dense matrices over Rational Field
sage: MS.ideal([MS.0,MS.1],side='right')
Right Ideal
(
  [1 0]
  [0 0],

  [0 1]
  [0 0]
)
 of Full MatrixSpace of 2 by 2 dense matrices over Rational Field
ideal_monoid()

The monoid of the ideals of this ring.

NOTE:

The code is copied from the base class of rings. This is since there are rings that do not inherit from that class, such as matrix algebras. See trac ticket #7797.

EXAMPLES:

sage: MS = MatrixSpace(QQ,2,2)
sage: isinstance(MS,Ring)
False
sage: MS in Rings()
True
sage: MS.ideal_monoid()
Monoid of ideals of Full MatrixSpace of 2 by 2 dense matrices
over Rational Field

Note that the monoid is cached:

sage: MS.ideal_monoid() is MS.ideal_monoid()
True
is_ring()

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

EXAMPLES:

sage: Parent(QQ,category=Rings()).is_ring()
True
is_zero()

Return True if this is the zero ring.

EXAMPLES:

sage: Integers(1).is_zero()
True
sage: Integers(2).is_zero()
False
sage: QQ.is_zero()
False
sage: R.<x> = ZZ[]
sage: R.quo(1).is_zero()
True
sage: R.<x> = GF(101)[]
sage: R.quo(77).is_zero()
True
sage: R.quo(x^2+1).is_zero()
False
quo(I, names=None)

Quotient of a ring by a two-sided ideal.

NOTE:

This is a synonym for quotient().

EXAMPLES:

sage: MS = MatrixSpace(QQ,2)
sage: I = MS*MS.gens()*MS

MS is not an instance of Ring.

However it is an instance of the parent class of the category of rings. The quotient method is inherited from there:

sage: isinstance(MS,sage.rings.ring.Ring)
False
sage: isinstance(MS,Rings().parent_class)
True
sage: MS.quo(I,names = ['a','b','c','d'])
Quotient of Full MatrixSpace of 2 by 2 dense matrices over Rational Field by the ideal
(
  [1 0]
  [0 0],

  [0 1]
  [0 0],

  [0 0]
  [1 0],

  [0 0]
  [0 1]
)
quotient(I, names=None)

Quotient of a ring by a two-sided ideal.

INPUT:

  • I: A twosided ideal of this ring.
  • names: a list of strings to be used as names for the variables in the quotient ring.

EXAMPLES:

Usually, a ring inherits a method sage.rings.ring.Ring.quotient(). So, we need a bit of effort to make the following example work with the category framework:

sage: F.<x,y,z> = FreeAlgebra(QQ)
sage: from sage.rings.noncommutative_ideals import Ideal_nc
sage: from itertools import product
sage: class PowerIdeal(Ideal_nc):
....:  def __init__(self, R, n):
....:      self._power = n
....:      Ideal_nc.__init__(self, R, [R.prod(m) for m in product(R.gens(), repeat=n)])
....:  def reduce(self, x):
....:      R = self.ring()
....:      return add([c*R(m) for m,c in x if len(m) < self._power], R(0))
....:
sage: I = PowerIdeal(F,3)
sage: Q = Rings().parent_class.quotient(F, I); Q
Quotient of Free Algebra on 3 generators (x, y, z) over Rational Field by the ideal (x^3, x^2*y, x^2*z, x*y*x, x*y^2, x*y*z, x*z*x, x*z*y, x*z^2, y*x^2, y*x*y, y*x*z, y^2*x, y^3, y^2*z, y*z*x, y*z*y, y*z^2, z*x^2, z*x*y, z*x*z, z*y*x, z*y^2, z*y*z, z^2*x, z^2*y, z^3)
sage: Q.0
xbar
sage: Q.1
ybar
sage: Q.2
zbar
sage: Q.0*Q.1
xbar*ybar
sage: Q.0*Q.1*Q.0
0
quotient_ring(I, names=None)

Quotient of a ring by a two-sided ideal.

NOTE:

This is a synonyme for quotient().

EXAMPLES:

sage: MS = MatrixSpace(QQ,2)
sage: I = MS*MS.gens()*MS

MS is not an instance of Ring, but it is an instance of the parent class of the category of rings. The quotient method is inherited from there:

sage: isinstance(MS,sage.rings.ring.Ring)
False
sage: isinstance(MS,Rings().parent_class)
True
sage: MS.quotient_ring(I,names = ['a','b','c','d'])
Quotient of Full MatrixSpace of 2 by 2 dense matrices over Rational Field by the ideal
(
  [1 0]
  [0 0],

  [0 1]
  [0 0],

  [0 0]
  [1 0],

  [0 0]
  [0 1]
)
class SubcategoryMethods
Division()

Return the full subcategory of the division objects of self.

A ring satisfies the division axiom if all non-zero elements have multiplicative inverses.

Note

This could be generalized to MagmasAndAdditiveMagmas.Distributive.AdditiveUnital.

EXAMPLES:

sage: Rings().Division()
Category of division rings
sage: Rings().Commutative().Division()
Category of fields
NoZeroDivisors()

Return the full subcategory of the objects of self having no nonzero zero divisors.

A zero divisor in a ring \(R\) is an element \(x \in R\) such that there exists a nonzero element \(y \in R\) such that \(x \cdot y = 0\) or \(y \cdot x = 0\) (see Wikipedia article Zero_divisor).

EXAMPLES:

sage: Rings().NoZeroDivisors()
Category of domains

Note

This could be generalized to MagmasAndAdditiveMagmas.Distributive.AdditiveUnital.