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119 lines
3.2 KiB
119 lines
3.2 KiB
5 months ago
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"""Implementation of :class:`Ring` class. """
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from sympy.polys.domains.domain import Domain
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from sympy.polys.polyerrors import ExactQuotientFailed, NotInvertible, NotReversible
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from sympy.utilities import public
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@public
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class Ring(Domain):
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"""Represents a ring domain. """
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is_Ring = True
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def get_ring(self):
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"""Returns a ring associated with ``self``. """
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return self
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def exquo(self, a, b):
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"""Exact quotient of ``a`` and ``b``, implies ``__floordiv__``. """
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if a % b:
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raise ExactQuotientFailed(a, b, self)
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else:
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return a // b
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def quo(self, a, b):
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"""Quotient of ``a`` and ``b``, implies ``__floordiv__``. """
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return a // b
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def rem(self, a, b):
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"""Remainder of ``a`` and ``b``, implies ``__mod__``. """
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return a % b
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def div(self, a, b):
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"""Division of ``a`` and ``b``, implies ``__divmod__``. """
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return divmod(a, b)
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def invert(self, a, b):
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"""Returns inversion of ``a mod b``. """
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s, t, h = self.gcdex(a, b)
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if self.is_one(h):
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return s % b
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else:
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raise NotInvertible("zero divisor")
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def revert(self, a):
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"""Returns ``a**(-1)`` if possible. """
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if self.is_one(a) or self.is_one(-a):
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return a
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else:
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raise NotReversible('only units are reversible in a ring')
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def is_unit(self, a):
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try:
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self.revert(a)
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return True
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except NotReversible:
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return False
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def numer(self, a):
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"""Returns numerator of ``a``. """
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return a
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def denom(self, a):
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"""Returns denominator of `a`. """
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return self.one
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def free_module(self, rank):
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"""
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Generate a free module of rank ``rank`` over self.
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>>> from sympy.abc import x
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>>> from sympy import QQ
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>>> QQ.old_poly_ring(x).free_module(2)
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QQ[x]**2
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"""
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raise NotImplementedError
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def ideal(self, *gens):
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"""
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Generate an ideal of ``self``.
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>>> from sympy.abc import x
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>>> from sympy import QQ
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>>> QQ.old_poly_ring(x).ideal(x**2)
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<x**2>
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"""
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from sympy.polys.agca.ideals import ModuleImplementedIdeal
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return ModuleImplementedIdeal(self, self.free_module(1).submodule(
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*[[x] for x in gens]))
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def quotient_ring(self, e):
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"""
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Form a quotient ring of ``self``.
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Here ``e`` can be an ideal or an iterable.
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>>> from sympy.abc import x
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>>> from sympy import QQ
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>>> QQ.old_poly_ring(x).quotient_ring(QQ.old_poly_ring(x).ideal(x**2))
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QQ[x]/<x**2>
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>>> QQ.old_poly_ring(x).quotient_ring([x**2])
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QQ[x]/<x**2>
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The division operator has been overloaded for this:
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>>> QQ.old_poly_ring(x)/[x**2]
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QQ[x]/<x**2>
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"""
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from sympy.polys.agca.ideals import Ideal
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from sympy.polys.domains.quotientring import QuotientRing
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if not isinstance(e, Ideal):
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e = self.ideal(*e)
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return QuotientRing(self, e)
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def __truediv__(self, e):
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return self.quotient_ring(e)
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