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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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