def __init__(self,
base,
p,
prec,
print_mode,
names,
element_class,
category=None):
"""
Initialization.
INPUT:
- base -- Base ring.
- p -- prime
- print_mode -- dictionary of print options
- names -- how to print the uniformizer
- element_class -- the class for elements of this ring
EXAMPLES::
sage: R = Zp(17) #indirect doctest
"""
if category is None:
if self.is_field():
category = Fields()
else:
category = PrincipalIdealDomains()
category = category.Metric().Complete()
LocalGeneric.__init__(self, base, prec, names, element_class, category)
self._printer = pAdicPrinter(self, print_mode)
def super_categories(self):
"""
EXAMPLES::
sage: EuclideanDomains().super_categories()
[Category of principal ideal domains]
"""
return [PrincipalIdealDomains()]
def super_categories(self):
"""
EXAMPLES::
sage: DiscreteValuationRings().super_categories()
[Category of principal ideal domains]
"""
return [PrincipalIdealDomains()]
def ideal(self, *args, **kwds):
"""
Create an ideal of this ring.
NOTE:
The code is copied from the base class
:class:`~sage.rings.ring.Ring`. This is
because there are rings that do not inherit
from that class, such as matrix algebras.
See :trac:`7797`.
INPUT:
- An element or a list/tuple/sequence of elements.
- ``coerce`` (optional bool, default ``True``):
First coerce the elements into this ring.
- ``side``, optional string, one of ``"twosided"``
(default), ``"left"``, ``"right"``: determines
whether the resulting ideal is twosided, a left
ideal or a right ideal.
EXAMPLE::
sage: MS = MatrixSpace(QQ,2,2)
sage: isinstance(MS,Ring)
False
sage: MS in Rings()
True
sage: MS.ideal(2)
Twosided Ideal
(
[2 0]
[0 2]
)
of Full MatrixSpace of 2 by 2 dense matrices over Rational Field
sage: MS.ideal([MS.0,MS.1],side='right')
Right Ideal
(
[1 0]
[0 0],
<BLANKLINE>
[0 1]
[0 0]
)
of Full MatrixSpace of 2 by 2 dense matrices over Rational Field
"""
if kwds.has_key('coerce'):
coerce = kwds['coerce']
del kwds['coerce']
else:
coerce = True
from sage.rings.ideal import Ideal_generic
from types import GeneratorType
if len(args) == 0:
gens = [self(0)]
else:
gens = args
while isinstance(
gens, (list, tuple, GeneratorType)) and len(gens) == 1:
first = gens[0]
if isinstance(first, Ideal_generic):
R = first.ring()
m = self.convert_map_from(R)
if m is not None:
gens = [m(g) for g in first.gens()]
coerce = False
else:
m = R.convert_map_from(self)
if m is not None:
raise NotImplementedError
else:
raise TypeError
break
elif isinstance(first, (list, tuple, GeneratorType)):
gens = first
else:
try:
if self.has_coerce_map_from(first):
gens = first.gens(
) # we have a ring as argument
elif hasattr(first, 'parent'):
gens = [first]
else:
raise ArithmeticError, "There is no coercion from %s to %s" % (
first, self)
except TypeError: # first may be a ring element
pass
break
if coerce:
gens = [self(g) for g in gens]
from sage.categories.principal_ideal_domains import PrincipalIdealDomains
if self in PrincipalIdealDomains():
# Use GCD algorithm to obtain a principal ideal
g = gens[0]
if len(gens) == 1:
try:
g = g.gcd(
g
) # note: we set g = gcd(g, g) to "canonicalize" the generator: make polynomials monic, etc.
except (AttributeError, NotImplementedError):
pass
else:
for h in gens[1:]:
g = g.gcd(h)
gens = [g]
if kwds.has_key('ideal_class'):
C = kwds['ideal_class']
del kwds['ideal_class']
else:
C = self._ideal_class_(len(gens))
if len(gens) == 1 and isinstance(gens[0], (list, tuple)):
gens = gens[0]
return C(self, gens, **kwds)