Rings#

class sage.categories.rings.Rings(base_category)#

Bases: 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 github issue #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 CommutativeRings

Division#

alias of DivisionRings

class ElementMethods#

Bases: object

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)                                            # needs sage.libs.pari
sage: S(1).inverse_of_unit()                                            # needs sage.libs.pari
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: # needs sage.modules
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#

Bases: object

extend_to_fraction_field()#

Return the extension of this morphism to fraction fields of the domain and the codomain.

EXAMPLES:

sage: S.<x> = QQ[]
sage: f = S.hom([x + 1]); f
Ring endomorphism of Univariate Polynomial Ring in x over Rational Field
    Defn: x |--> x + 1

sage: g = f.extend_to_fraction_field(); g                               # needs sage.libs.singular
Ring endomorphism of Fraction Field of Univariate Polynomial Ring in x
 over Rational Field
    Defn: x |--> x + 1
sage: g(x)                                                              # needs sage.libs.singular
x + 1
sage: g(1/x)                                                            # needs sage.libs.singular
1/(x + 1)

If this morphism is not injective, it does not extend to the fraction field and an error is raised:

sage: f = GF(5).coerce_map_from(ZZ)
sage: f.extend_to_fraction_field()
Traceback (most recent call last):
...
ValueError: the morphism is not injective
is_injective()#

Return whether or not this morphism is injective.

EXAMPLES:

sage: # needs sage.libs.singular
sage: R.<x,y> = QQ[]
sage: R.hom([x, y^2], R).is_injective()
True
sage: R.hom([x, x^2], R).is_injective()
False
sage: S.<u,v> = R.quotient(x^3*y)
sage: R.hom([v, u], S).is_injective()
False
sage: S.hom([-u, v], S).is_injective()
True
sage: S.cover().is_injective()
False

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)                                                       # needs sage.rings.padics
sage: R.fraction_field().coerce_map_from(R).is_injective()
True
NoZeroDivisors#

alias of Domains

class ParentMethods#

Bases: object

bracket(x, y)#

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

INPUT:

  • x, y – elements of self

EXAMPLES:

sage: # needs sage.combinat sage.modules
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)      # needs sage.combinat sage.modules
....:  + 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()                                            # needs sage.rings.padics
0
free_module(base=None, basis=None, map=True)#

Return a free module \(V\) over the specified subring together with maps to and from \(V\).

The default implementation only supports the case that the base ring is the ring itself.

INPUT:

  • base – a subring \(R\) so that this ring is isomorphic to a finite-rank free \(R\)-module \(V\)

  • basis – (optional) a basis for this ring over the base

  • map – boolean (default True), whether to return \(R\)-linear maps to and from \(V\)

OUTPUT:

  • A finite-rank free \(R\)-module \(V\)

  • An \(R\)-module isomorphism from \(V\) to this ring (only included if map is True)

  • An \(R\)-module isomorphism from this ring to \(V\) (only included if map is True)

EXAMPLES:

sage: # needs sage.modules
sage: R.<x> = QQ[[]]
sage: V, from_V, to_V = R.free_module(R)
sage: v = to_V(1 + x); v
(1 + x)
sage: from_V(v)
1 + x
sage: W, from_W, to_W = R.free_module(R, basis=(1 - x))
sage: W is V
True
sage: w = to_W(1 + x); w
(1 - x^2)
sage: from_W(w)
1 + x + O(x^20)
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 github issue #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: # needs sage.modules
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 github issue #7797.

EXAMPLES:

sage: # needs sage.modules
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()                            # needs sage.modules
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()                                          # needs sage.libs.pari
False
quo(I, names=None, **kwds)#

Quotient of a ring by a two-sided ideal.

Note

This is a synonym for quotient().

EXAMPLES:

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

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)                              # needs sage.modules
False
sage: isinstance(MS, Rings().parent_class)                              # needs sage.modules
True
sage: MS.quo(I, names=['a','b','c','d'])                                # needs sage.modules
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]
)

A test with a subclass of Ring:

sage: # needs sage.libs.singular
sage: R.<x,y> = PolynomialRing(QQ, 2)
sage: S.<a,b> = R.quo((x^2, y))
sage: S
Quotient of Multivariate Polynomial Ring in x, y over Rational Field
 by the ideal (x^2, y)
sage: S.gens()
(a, 0)
sage: a == b
False
quotient(I, names=None, **kwds)#

Quotient of a ring by a two-sided ideal.

INPUT:

  • I – A twosided ideal of this ring.

  • names – (optional) names of the generators of the quotient (if there are multiple generators, you can specify a single character string and the generators are named in sequence starting with 0).

  • further named arguments that may be passed to the quotient ring constructor.

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: # needs sage.combinat sage.modules
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

An example with polynomial rings:

sage: R.<x> = PolynomialRing(ZZ)
sage: I = R.ideal([4 + 3*x + x^2, 1 + x^2])
sage: S = R.quotient(I, 'a')
sage: S.gens()
(a,)

sage: # needs sage.libs.singular
sage: R.<x,y> = PolynomialRing(QQ, 2)
sage: S.<a,b> = R.quotient((x^2, y))
sage: S
Quotient of Multivariate Polynomial Ring in x, y over Rational Field
 by the ideal (x^2, y)
sage: S.gens()
(a, 0)
sage: a == b
False
quotient_ring(I, names=None, **kwds)#

Quotient of a ring by a two-sided ideal.

Note

This is a synonym for quotient().

INPUT:

  • I – an ideal of \(R\)

  • names – (optional) names of the generators of the quotient. (If there are multiple generators, you can specify a single character string and the generators are named in sequence starting with 0.)

  • further named arguments that may be passed to the quotient ring constructor.

OUTPUT:

  • R/I – the quotient ring of \(R\) by the ideal \(I\)

EXAMPLES:

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

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)                              # needs sage.modules
False
sage: isinstance(MS, Rings().parent_class)                              # needs sage.modules
True
sage: MS.quotient_ring(I, names=['a','b','c','d'])                      # needs sage.modules
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]
)

A test with a subclass of Ring:

sage: R.<x> = PolynomialRing(ZZ)
sage: I = R.ideal([4 + 3*x + x^2, 1 + x^2])
sage: S = R.quotient_ring(I, 'a')
sage: S.gens()
(a,)

sage: # needs sage.libs.singular
sage: R.<x,y> = PolynomialRing(QQ,2)
sage: S.<a,b> = R.quotient_ring((x^2, y))
sage: S
Quotient of Multivariate Polynomial Ring in x, y over Rational Field
 by the ideal (x^2, y)
sage: S.gens()
(a, 0)
sage: a == b
False
class SubcategoryMethods#

Bases: object

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.

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