def construction(self):
"""
EXAMPLES::
sage: R.<x> = PolynomialRing(ZZ,'x')
sage: I = R.ideal([4 + 3*x + x^2, 1 + x^2])
sage: R.quotient_ring(I).construction()
(QuotientFunctor, Univariate Polynomial Ring in x over Integer Ring)
TESTS::
sage: F, R = Integers(5).construction()
sage: F(R)
Ring of integers modulo 5
sage: F, R = GF(5).construction()
sage: F(R)
Finite Field of size 5
"""
from sage.categories.pushout import QuotientFunctor
# Is there a better generic way to distinguish between things like Z/pZ as a field and Z/pZ as a ring?
from sage.rings.field import Field
try:
names = self.variable_names()
except ValueError:
try:
names = self.cover_ring().variable_names()
except ValueError:
names = None
return QuotientFunctor(self.__I,
names=names,
as_field=isinstance(self, Field)), self.__R
def construction(self):
"""
Returns the functorial construction of ``self``.
EXAMPLES::
sage: R.<x> = PolynomialRing(ZZ,'x')
sage: I = R.ideal([4 + 3*x + x^2, 1 + x^2])
sage: R.quotient_ring(I).construction()
(QuotientFunctor, Univariate Polynomial Ring in x over Integer Ring)
sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace')
sage: I = F*[x*y+y*z,x^2+x*y-y*x-y^2]*F
sage: Q = F.quo(I)
sage: Q.construction()
(QuotientFunctor, Free Associative Unital Algebra on 3 generators (x, y, z) over Rational Field)
TESTS::
sage: F, R = Integers(5).construction()
sage: F(R)
Ring of integers modulo 5
sage: F, R = GF(5).construction()
sage: F(R)
Finite Field of size 5
"""
from sage.categories.pushout import QuotientFunctor
# Is there a better generic way to distinguish between things like Z/pZ as a field and Z/pZ as a ring?
from sage.rings.ring import Field
try:
names = self.variable_names()
except ValueError:
try:
names = self.cover_ring().variable_names()
except ValueError:
names = None
if self in CommutativeRings():
return QuotientFunctor(self.__I,
names=names,
domain=CommutativeRings(),
codomain=CommutativeRings(),
as_field=isinstance(self, Field)), self.__R
else:
return QuotientFunctor(self.__I,
names=names,
as_field=isinstance(self, Field)), self.__R