def __init__(self, R, I, names, category=None):
"""
Create the quotient ring of `R` by the twosided ideal `I`.
INPUT:
- ``R`` -- a ring.
- ``I`` -- a twosided ideal of `R`.
- ``names`` -- a list of generator names.
EXAMPLES::
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.<a,b,c> = F.quo(I); Q
Quotient of Free Associative Unital Algebra on 3 generators (x, y, z) over Rational Field by the ideal (x*y + y*z, x*x + x*y - y*x - y*y)
sage: a*b
-b*c
sage: a^3
-b*c*a - b*c*b - b*c*c
"""
if R not in _Rings:
raise TypeError(
"The first argument must be a ring, but %s is not" % R)
if I not in R.ideal_monoid():
raise TypeError(
"The second argument must be an ideal of the given ring, but %s is not"
% I)
self.__R = R
self.__I = I
#sage.structure.parent_gens.ParentWithGens.__init__(self, R.base_ring(), names)
##
# Unfortunately, computing the join of categories, which is done in
# check_default_category, is very expensive.
# However, we don't just want to use the given category without mixing in
# some quotient stuff - unless Parent.__init__ was called
# previously, in which case the quotient ring stuff is just
# a vaste of time. This is the case for FiniteField_prime_modn.
if not self._is_category_initialized():
if category is None:
try:
commutative = R.is_commutative()
except (AttributeError, NotImplementedError):
commutative = False
if commutative:
category = check_default_category(
_CommutativeRingsQuotients, category)
else:
category = check_default_category(_RingsQuotients,
category)
ring.Ring.__init__(self,
R.base_ring(),
names=names,
category=category)
def __init__(self, R, I, names, category=None):
"""
Create the quotient ring of `R` by the twosided ideal `I`.
INPUT:
- ``R`` -- a ring.
- ``I`` -- a twosided ideal of `R`.
- ``names`` -- a list of generator names.
EXAMPLES::
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.<a,b,c> = F.quo(I); Q
Quotient of Free Associative Unital Algebra on 3 generators (x, y, z) over Rational Field by the ideal (x*y + y*z, x*x + x*y - y*x - y*y)
sage: a*b
-b*c
sage: a^3
-b*c*a - b*c*b - b*c*c
"""
if R not in _Rings:
raise TypeError("The first argument must be a ring, but %s is not"%R)
if I not in R.ideal_monoid():
raise TypeError("The second argument must be an ideal of the given ring, but %s is not"%I)
self.__R = R
self.__I = I
#sage.structure.parent_gens.ParentWithGens.__init__(self, R.base_ring(), names)
##
# Unfortunately, computing the join of categories, which is done in
# check_default_category, is very expensive.
# However, we don't just want to use the given category without mixing in
# some quotient stuff - unless Parent.__init__ was called
# previously, in which case the quotient ring stuff is just
# a vaste of time. This is the case for FiniteField_prime_modn.
if not self._is_category_initialized():
if category is None:
try:
commutative = R.is_commutative()
except (AttributeError, NotImplementedError):
commutative = False
if commutative:
category = check_default_category(_CommutativeRingsQuotients,category)
else:
category = check_default_category(_RingsQuotients,category)
ring.Ring.__init__(self, R.base_ring(), names=names, category=category)
def __init__(self, R, I, names, category=None):
"""
Initialize ``self``.
INPUT:
- ``R`` -- a ring that is a :class:`~sage.rings.ring.CommutativeRing`.
- ``I`` -- an ideal of `R`.
- ``names`` -- a list of generator names.
TESTS::
sage: isinstance(ZZ.quo(2), sage.rings.ring.CommutativeRing) # indirect doctest
True
"""
if not isinstance(R, ring.CommutativeRing):
raise TypeError(
"This class is for quotients of commutative rings only.\n For non-commutative rings, use <sage.rings.quotient_ring.QuotientRing_nc>"
)
if not self._is_category_initialized():
category = check_default_category(_CommutativeRingsQuotients,
category)
QuotientRing_nc.__init__(self, R, I, names, category=category)
def __init__(self, base, prec, names, element_class, category=None):
"""
Initializes self.
EXAMPLES::
sage: R = Zp(5) #indirect doctest
sage: R.precision_cap()
20
In :trac:`14084`, the category framework has been implemented for p-adic rings::
sage: TestSuite(R).run()
sage: K = Qp(7)
sage: TestSuite(K).run()
TESTS::
sage: R = Zp(5, 5, 'fixed-mod')
sage: R._repr_option('element_is_atomic')
False
"""
self._prec = prec
self.Element = element_class
default_category = getattr(self, '_default_category', None)
if self.is_field():
category = CompleteDiscreteValuationFields()
else:
category = CompleteDiscreteValuationRings()
if default_category is not None:
category = check_default_category(default_category, category)
Parent.__init__(self, base, names=(names,), normalize=False, category=category, element_constructor=element_class)
def __init__(self, base, prec, names, element_class, category=None):
"""
Initializes self.
EXAMPLES::
sage: R = Zp(5) #indirect doctest
sage: R.precision_cap()
20
In :trac:`14084`, the category framework has been implemented for p-adic rings::
sage: TestSuite(R).run()
sage: K = Qp(7)
sage: TestSuite(K).run()
TESTS::
sage: R = Zp(5, 5, 'fixed-mod')
sage: R._repr_option('element_is_atomic')
False
"""
self._prec = prec
self.Element = element_class
default_category = getattr(self, '_default_category', None)
if self.is_field():
category = CompleteDiscreteValuationFields()
else:
category = CompleteDiscreteValuationRings()
if default_category is not None:
category = check_default_category(default_category, category)
Parent.__init__(self, base, names=(names,), normalize=False, category=category, element_constructor=element_class)
def __init__(self, R, I, names, category=None):
"""
Create the quotient ring of `R` by the twosided ideal `I`.
INPUT:
- ``R`` - a ring.
- ``I`` - a twosided ideal of `R`.
- ``names`` - a list of generator names.
"""
if R not in _Rings:
raise TypeError, "The first argument must be a ring, but %s is not" % R
if I not in R.ideal_monoid():
raise TypeError, "The second argument must be an ideal of the given ring, but %s is not" % I
self.__R = R
self.__I = I
#sage.structure.parent_gens.ParentWithGens.__init__(self, R.base_ring(), names)
##
# Unfortunately, computing the join of categories, which is done in
# check_default_category, is very expensive.
# However, we don't just want to use the given category without mixing in
# some quotient stuff - unless Parent.__init__ was called
# previously, in which case the quotient ring stuff is just
# a vaste of time. This is the case for FiniteField_prime_modn.
if not self._is_category_initialized():
if category is None:
try:
commutative = R.is_commutative()
except (AttributeError, NotImplementedError):
commutative = False
if commutative:
category = check_default_category(
_CommutativeRingsQuotients, category)
else:
category = check_default_category(_RingsQuotients,
category)
ring.Ring.__init__(self,
R.base_ring(),
names=names,
category=category)
def __init__(self, R, I, names, category=None):
"""
Create the quotient ring of `R` by the twosided ideal `I`.
INPUT:
- ``R`` - a ring.
- ``I`` - a twosided ideal of `R`.
- ``names`` - a list of generator names.
"""
if R not in _Rings:
raise TypeError, "The first argument must be a ring, but %s is not"%R
if I not in R.ideal_monoid():
raise TypeError, "The second argument must be an ideal of the given ring, but %s is not"%I
self.__R = R
self.__I = I
#sage.structure.parent_gens.ParentWithGens.__init__(self, R.base_ring(), names)
##
# Unfortunately, computing the join of categories, which is done in
# check_default_category, is very expensive.
# However, we don't just want to use the given category without mixing in
# some quotient stuff - unless Parent.__init__ was called
# previously, in which case the quotient ring stuff is just
# a vaste of time. This is the case for FiniteField_prime_modn.
if not self._is_category_initialized():
if category is None:
try:
commutative = R.is_commutative()
except (AttributeError, NotImplementedError):
commutative = False
if commutative:
category = check_default_category(_CommutativeRingsQuotients,category)
else:
category = check_default_category(_RingsQuotients,category)
ring.Ring.__init__(self, R.base_ring(), names=names, category=category)
def __init__(self, R, I, names, category=None):
"""
INPUT:
- ``R`` - a ring that is of type <sage.rings.ring.CommutativeRing>.
- ``I`` - an ideal of `R`.
- ``names`` - a list of generator names.
TEST::
sage: isinstance(ZZ.quo(2), sage.rings.ring.CommutativeRing) # indirect doctest
True
"""
if not isinstance(R, sage.rings.commutative_ring.CommutativeRing):
raise TypeError, "This class is for quotients of commutative rings only.\n For non-commutative rings, use <sage.rings.quotient_ring.QuotientRing_nc>"
if not self._is_category_initialized():
category = check_default_category(_CommutativeRingsQuotients,category)
QuotientRing_nc.__init__(self, R, I, names, category=category)
