"""
Monoid of ideals in a commutative ring
"""
from commutative_ring import is_CommutativeRing
from sage.structure.parent import Parent
import sage.rings.integer_ring
import ideal
from sage.categories.monoids import Monoids
def IdealMonoid(R):
r"""
Return the monoid of ideals in the ring ``R``.
EXAMPLE::
sage: R = QQ['x']
sage: sage.rings.ideal_monoid.IdealMonoid(R)
Monoid of ideals of Univariate Polynomial Ring in x over Rational Field
"""
return IdealMonoid_c(R)
class IdealMonoid_c(Parent):
r"""
The monoid of ideals in a commutative ring.
TESTS::
sage: R = QQ['x']
sage: M = sage.rings.ideal_monoid.IdealMonoid(R)
sage: TestSuite(M).run()
Failure in _test_category:
...
The following tests failed: _test_elements
(The "_test_category" test fails but I haven't the foggiest idea why.)
"""
Element = ideal.Ideal_generic
def __init__(self, R):
r"""
Initialize ``self``.
TESTS::
sage: R = QuadraticField(-23, 'a')
sage: M = sage.rings.ideal_monoid.IdealMonoid(R); M # indirect doctest
Monoid of ideals of Number Field in a with defining polynomial x^2 + 23
"""
self.__R = R
Parent.__init__(self, base = sage.rings.integer_ring.ZZ, category = Monoids())
self._populate_coercion_lists_()
def _repr_(self):
r"""
Return a string representation of ``self``.
TESTS::
sage: R = QuadraticField(-23, 'a')
sage: M = sage.rings.ideal_monoid.IdealMonoid(R); M._repr_()
'Monoid of ideals of Number Field in a with defining polynomial x^2 + 23'
"""
return "Monoid of ideals of %s"%self.__R
def ring(self):
r"""
Return the ring of which this is the ideal monoid.
EXAMPLE::
sage: R = QuadraticField(-23, 'a')
sage: M = sage.rings.ideal_monoid.IdealMonoid(R); M.ring() is R
True
"""
return self.__R
def _element_constructor_(self, x):
r"""
Create an ideal in this monoid from ``x``.
EXAMPLES::
sage: R.<a> = QuadraticField(-23)
sage: M = sage.rings.ideal_monoid.IdealMonoid(R)
sage: M(a) # indirect doctest
Fractional ideal (a)
sage: M([a-4, 13])
Fractional ideal (13, 1/2*a + 9/2)
"""
try:
side = x.side()
except (AttributeError,TypeError):
side = None
try:
x = x.gens()
except AttributeError:
pass
if side is None:
y = self.__R.ideal(x)
else:
y = self.__R.ideal(x,side=side)
y._set_parent(self)
return y
def _coerce_map_from_(self, x):
r"""
Used by coercion framework.
EXAMPLE::
sage: R = QuadraticField(-23, 'a')
sage: M = R.ideal_monoid()
sage: M.has_coerce_map_from(R) # indirect doctest
True
sage: M.has_coerce_map_from(QQ.ideal_monoid())
True
sage: M.has_coerce_map_from(Zmod(6))
False
sage: M.has_coerce_map_from(loads(dumps(M)))
True
"""
if isinstance(x, IdealMonoid_c):
return self.ring().has_coerce_map_from(x.ring())
else:
return self.ring().has_coerce_map_from(x)
def __cmp__(self, other):
r"""
Comparison function.
EXAMPLE::
sage: R = QuadraticField(-23, 'a')
sage: M = R.ideal_monoid()
sage: M == QQ
False
sage: M == 17
False
sage: M == R.ideal_monoid()
True
"""
if not isinstance(other, IdealMonoid_c):
return cmp(type(self), type(other))
else:
return cmp(self.ring(), other.ring())
