Source code for sympy.polys.domains.quotientring

"""Implementation of :class:`QuotientRing` class."""

from __future__ import print_function, division

from sympy.polys.domains.ring import Ring
from sympy.polys.polyerrors import NotReversible, CoercionFailed
from sympy.polys.agca.modules import FreeModuleQuotientRing

from sympy.utilities import public

# TODO
# - successive quotients (when quotient ideals are implemented)
# - poly rings over quotients?
# - division by non-units in integral domains?

@public
class QuotientRingElement(object):
    """
    Class representing elements of (commutative) quotient rings.

    Attributes:

    - ring - containing ring
    - data - element of ring.ring (i.e. base ring) representing self
    """

    def __init__(self, ring, data):
        self.ring = ring
        self.data = data

    def __str__(self):
        from sympy import sstr
        return sstr(self.data) + " + " + str(self.ring.base_ideal)

    def __add__(self, om):
        if not isinstance(om, self.__class__) or om.ring != self.ring:
            try:
                om = self.ring.convert(om)
            except (NotImplementedError, CoercionFailed):
                return NotImplemented
        return self.ring(self.data + om.data)

    __radd__ = __add__

    def __neg__(self):
        return self.ring(self.data*self.ring.ring.convert(-1))

    def __sub__(self, om):
        return self.__add__(-om)

    def __rsub__(self, om):
        return (-self).__add__(om)

    def __mul__(self, o):
        if not isinstance(o, self.__class__):
            try:
                o = self.ring.convert(o)
            except (NotImplementedError, CoercionFailed):
                return NotImplemented
        return self.ring(self.data*o.data)

    __rmul__ = __mul__

    def __rdiv__(self, o):
        return self.ring.revert(self)*o

    __rtruediv__ = __rdiv__

    def __div__(self, o):
        if not isinstance(o, self.__class__):
            try:
                o = self.ring.convert(o)
            except (NotImplementedError, CoercionFailed):
                return NotImplemented
        return self.ring.revert(o)*self

    __truediv__ = __div__

    def __pow__(self, oth):
        return self.ring(self.data**oth)

    def __eq__(self, om):
        if not isinstance(om, self.__class__) or om.ring != self.ring:
            return False
        return self.ring.is_zero(self - om)

    def __ne__(self, om):
        return not self == om


[docs]class QuotientRing(Ring): """ Class representing (commutative) quotient rings. You should not usually instantiate this by hand, instead use the constructor from the base ring in the construction. >>> from sympy.abc import x >>> from sympy import QQ >>> I = QQ.old_poly_ring(x).ideal(x**3 + 1) >>> QQ.old_poly_ring(x).quotient_ring(I) QQ[x]/<x**3 + 1> Shorter versions are possible: >>> QQ.old_poly_ring(x)/I QQ[x]/<x**3 + 1> >>> QQ.old_poly_ring(x)/[x**3 + 1] QQ[x]/<x**3 + 1> Attributes: - ring - the base ring - base_ideal - the ideal used to form the quotient """ has_assoc_Ring = True has_assoc_Field = False dtype = QuotientRingElement def __init__(self, ring, ideal): if not ideal.ring == ring: raise ValueError('Ideal must belong to %s, got %s' % (ring, ideal)) self.ring = ring self.base_ideal = ideal self.zero = self(self.ring.zero) self.one = self(self.ring.one) def __str__(self): return str(self.ring) + "/" + str(self.base_ideal) def __hash__(self): return hash((self.__class__.__name__, self.dtype, self.ring, self.base_ideal)) def new(self, a): """Construct an element of `self` domain from `a`. """ if not isinstance(a, self.ring.dtype): a = self.ring(a) # TODO optionally disable reduction? return self.dtype(self, self.base_ideal.reduce_element(a)) def __eq__(self, other): """Returns `True` if two domains are equivalent. """ return isinstance(other, QuotientRing) and \ self.ring == other.ring and self.base_ideal == other.base_ideal def from_ZZ_python(K1, a, K0): """Convert a Python `int` object to `dtype`. """ return K1(K1.ring.convert(a, K0)) from_QQ_python = from_ZZ_python from_ZZ_gmpy = from_ZZ_python from_QQ_gmpy = from_ZZ_python from_RealField = from_ZZ_python from_GlobalPolynomialRing = from_ZZ_python from_FractionField = from_ZZ_python def from_sympy(self, a): return self(self.ring.from_sympy(a)) def to_sympy(self, a): return self.ring.to_sympy(a.data) def from_QuotientRing(self, a, K0): if K0 == self: return a def poly_ring(self, *gens): """Returns a polynomial ring, i.e. `K[X]`. """ raise NotImplementedError('nested domains not allowed') def frac_field(self, *gens): """Returns a fraction field, i.e. `K(X)`. """ raise NotImplementedError('nested domains not allowed') def revert(self, a): """ Compute a**(-1), if possible. """ I = self.ring.ideal(a.data) + self.base_ideal try: return self(I.in_terms_of_generators(1)[0]) except ValueError: # 1 not in I raise NotReversible('%s not a unit in %r' % (a, self)) def is_zero(self, a): return self.base_ideal.contains(a.data) def free_module(self, rank): """ Generate a free module of rank ``rank`` over ``self``. >>> from sympy.abc import x >>> from sympy import QQ >>> (QQ.old_poly_ring(x)/[x**2 + 1]).free_module(2) (QQ[x]/<x**2 + 1>)**2 """ return FreeModuleQuotientRing(self, rank)