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 categoryAlgebras(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
- 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 ofself
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 basemap
– boolean (defaultTrue
), 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
isTrue
)An \(R\)-module isomorphism from this ring to \(V\) (only included if
map
isTrue
)
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, 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: # 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 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) # 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 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) # 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