Bases: sage.categories.category_singleton.Category_singleton
The category of rings
Associative rings with unit, not necessarily commutative
EXAMPLES:
sage: Rings()
Category of rings
sage: Rings().super_categories()
[Category of rngs, Category of semirings]
TESTS:
sage: TestSuite(Rings()).run()
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).
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()
Traceback (most recent call last):
...
NotImplementedError
Bases: sage.categories.category.HomCategory
Initializes this HomCategory
EXAMPLES:
We need to skip one test, since the hierarchy of hom categories isn’t consistent yet:
sage: C = sage.categories.category.HomCategory(Rings()); C
Category of hom sets in Category of rings
sage: TestSuite(C).run(skip=['_test_category_graph'])
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
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
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:
EXAMPLE:
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
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.
EXAMPLE:
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
Return True, since this in an object of the category of rings.
EXAMPLES:
sage: Parent(QQ,category=Rings()).is_ring()
True
Quotient of a ring by a two-sided ideal.
NOTE:
This is a synonyme for quotient().
EXAMPLE:
sage: MS = MatrixSpace(QQ,2)
sage: MS.full_category_initialisation()
sage: I = MS*MS.gens()*MS
MS is not an instance of Ring. But since its category was fully initalised (which is not by default, by trac ticket #11900), 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 of a ring by a two-sided ideal.
INPUT:
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: class PowerIdeal(Ideal_nc):
... def __init__(self, R, n):
... self._power = n
... self._power = n
... Ideal_nc.__init__(self,R,[R.prod(m) for m in CartesianProduct(*[R.gens()]*n)])
... def reduce(self,x):
... R = self.ring()
... return add([c*R(m) for c,m 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 of a ring by a two-sided ideal.
NOTE:
This is a synonyme for quotient().
EXAMPLE:
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]
)
EXAMPLES:
sage: Rings().super_categories()
[Category of rngs, Category of semirings]