def _coerce_impl(self, x):
"""
Check to see if we can coerce ``x`` into a homomorphism with the
correct rings.
EXAMPLES::
sage: H = Hom(ZZ, QQ)
sage: phi = H([1])
sage: H2 = Hom(QQ, QQ)
sage: phi2 = H2(phi); phi2 # indirect doctest
Ring endomorphism of Rational Field
Defn: 1 |--> 1
sage: H(phi2) # indirect doctest
Ring morphism:
From: Integer Ring
To: Rational Field
Defn: 1 |--> 1
"""
if not isinstance(x, morphism.RingHomomorphism):
raise TypeError
if x.parent() is self:
return x
# Case 1: the parent fits
if x.parent() == self:
if isinstance(x, morphism.RingHomomorphism_im_gens):
return morphism.RingHomomorphism_im_gens(self, x.im_gens())
elif isinstance(x, morphism.RingHomomorphism_cover):
return morphism.RingHomomorphism_cover(self)
elif isinstance(x, morphism.RingHomomorphism_from_base):
return morphism.RingHomomorphism_from_base(
self, x.underlying_map())
# Case 2: unique extension via fraction field
try:
if isinstance(x, morphism.RingHomomorphism_im_gens) and x.domain(
).fraction_field().has_coerce_map_from(self.domain()):
return morphism.RingHomomorphism_im_gens(self, x.im_gens())
except StandardError:
pass
# Case 3: the homomorphism can be extended by coercion
try:
return x.extend_codomain(self.codomain()).extend_domain(
self.domain())
except StandardError:
pass
# Last resort, case 4: the homomorphism is induced from the base ring
if self.domain() == self.domain().base() or self.codomain(
) == self.codomain().base():
raise TypeError
try:
x = self.domain().base().Hom(self.codomain().base())(x)
return morphism.RingHomomorphism_from_base(self, x)
except StandardError:
raise TypeError
def cover(self):
r"""
The covering ring homomorphism `R \to R/I`, equipped with a
section.
EXAMPLES::
sage: R = ZZ.quo(3*ZZ)
sage: pi = R.cover()
sage: pi
Ring morphism:
From: Integer Ring
To: Ring of integers modulo 3
Defn: Natural quotient map
sage: pi(5)
2
sage: l = pi.lift()
EXAMPLES::
sage: R.<x,y> = PolynomialRing(QQ)
sage: Q = R.quo( (x^2,y^2) )
sage: pi = Q.cover()
sage: pi(x^3+y)
ybar
sage: l = pi.lift(x+y^3)
sage: l
x
sage: l = pi.lift(); l
Set-theoretic ring morphism:
From: Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x^2, y^2)
To: Multivariate Polynomial Ring in x, y over Rational Field
Defn: Choice of lifting map
sage: l(x+y^3)
x
"""
try:
return self.__cover
except AttributeError:
import morphism
pi = morphism.RingHomomorphism_cover(self.__R.Hom(self))
lift = self.lifting_map()
pi._set_lift(lift)
self.__cover = pi
return self.__cover
def _coerce_impl(self, x):
if not isinstance(x, morphism.RingHomomorphism):
raise TypeError
if x.parent() is self:
return x
# Case 1: the parent fits
if x.parent() == self:
if isinstance(x, morphism.RingHomomorphism_im_gens):
return morphism.RingHomomorphism_im_gens(self, x.im_gens())
elif isinstance(x, morphism.RingHomomorphism_cover):
return morphism.RingHomomorphism_cover(self)
elif isinstance(x, morphism.RingHomomorphism_from_base):
return morphism.RingHomomorphism_from_base(
self, x.underlying_map())
# Case 2: unique extension via fraction field
try:
if isinstance(x, morphism.RingHomomorphism_im_gens) and x.domain(
).fraction_field().has_coerce_map_from(self.domain()):
return morphism.RingHomomorphism_im_gens(self, x.im_gens())
except:
pass
# Case 3: the homomorphism can be extended by coercion
try:
return x.extend_codomain(self.codomain()).extend_domain(
self.domain())
except:
pass
# Last resort, case 4: the homomorphism is induced from the base ring
if self.domain() == self.domain().base() or self.codomain(
) == self.codomain().base():
raise TypeError
try:
x = self.domain().base().Hom(self.codomain().base())(x)
return morphism.RingHomomorphism_from_base(self, x)
except:
raise TypeError