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 categoryAlgebras(P)
.

Commutative
¶

Division
¶

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) 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 (noncommutative) 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
¶ 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 Ring endomorphism of Fraction Field of Univariate Polynomial Ring in x over Rational Field Defn: x > x + 1 sage: g(x) x + 1 sage: g(1/x) 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: 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) sage: R.fraction_field().coerce_map_from(R).is_injective() True


NoZeroDivisors
¶ alias of
sage.categories.domains.Domains

class
ParentMethods
¶ Bases:
object

bracket
(x, y)¶ Returns the Lie bracket \([x, y] = x y  y x\) of \(x\) and \(y\).
INPUT:
x
,y
– elements ofself
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

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 finiterank free \(R\)module \(V\)basis
– (optional) a basis for this ring over the basemap
– boolean (defaultTrue
), whether to return \(R\)linear maps to and from \(V\)
OUTPUT:
A finiterank free \(R\)module \(V\)
An \(R\)module isomorphism from \(V\) to this ring (only included if
map
isTrue
)An \(R\)module isomorphism from this ring to \(V\) (only included if
map
isTrue
)
EXAMPLES:
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=(1x)) 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 trac ticket #7797.INPUT:
An element or a list/tuple/sequence of elements.
coerce
(optional bool, defaultTrue
): 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, **kwds)¶ Quotient of a ring by a twosided 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 ofRing
.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, **kwds)¶ Quotient of a ring by a twosided 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.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: 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, **kwds)¶ Quotient of a ring by a twosided 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 ofRing
, 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
¶ Bases:
object

Division
()¶ Return the full subcategory of the division objects of
self
.A ring satisfies the division axiom if all nonzero 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
.
