def Ideal(*args, **kwds):
r"""
Create the ideal in ring with given generators.
There are some shorthand notations for creating an ideal, in
addition to using the :func:`Ideal` function:
- ``R.ideal(gens, coerce=True)``
- ``gens*R``
- ``R*gens``
INPUT:
- ``R`` - A ring (optional; if not given, will try to infer it from
``gens``)
- ``gens`` - list of elements generating the ideal
- ``coerce`` - bool (optional, default: ``True``);
whether ``gens`` need to be coerced into the ring.
OUTPUT: The ideal of ring generated by ``gens``.
EXAMPLES::
sage: R, x = PolynomialRing(ZZ, 'x').objgen()
sage: I = R.ideal([4 + 3*x + x^2, 1 + x^2])
sage: I
Ideal (x^2 + 3*x + 4, x^2 + 1) of Univariate Polynomial Ring in x over Integer Ring
sage: Ideal(R, [4 + 3*x + x^2, 1 + x^2])
Ideal (x^2 + 3*x + 4, x^2 + 1) of Univariate Polynomial Ring in x over Integer Ring
sage: Ideal((4 + 3*x + x^2, 1 + x^2))
Ideal (x^2 + 3*x + 4, x^2 + 1) of Univariate Polynomial Ring in x over Integer Ring
::
sage: ideal(x^2-2*x+1, x^2-1)
Ideal (x^2 - 2*x + 1, x^2 - 1) of Univariate Polynomial Ring in x over Integer Ring
sage: ideal([x^2-2*x+1, x^2-1])
Ideal (x^2 - 2*x + 1, x^2 - 1) of Univariate Polynomial Ring in x over Integer Ring
sage: l = [x^2-2*x+1, x^2-1]
sage: ideal(f^2 for f in l)
Ideal (x^4 - 4*x^3 + 6*x^2 - 4*x + 1, x^4 - 2*x^2 + 1) of
Univariate Polynomial Ring in x over Integer Ring
This example illustrates how Sage finds a common ambient ring for
the ideal, even though 1 is in the integers (in this case).
::
sage: R.<t> = ZZ['t']
sage: i = ideal(1,t,t^2)
sage: i
Ideal (1, t, t^2) of Univariate Polynomial Ring in t over Integer Ring
sage: ideal(1/2,t,t^2)
Principal ideal (1) of Univariate Polynomial Ring in t over Rational Field
This shows that the issues at :trac:`1104` are resolved::
sage: Ideal(3, 5)
Principal ideal (1) of Integer Ring
sage: Ideal(ZZ, 3, 5)
Principal ideal (1) of Integer Ring
sage: Ideal(2, 4, 6)
Principal ideal (2) of Integer Ring
You have to provide enough information that Sage can figure out
which ring to put the ideal in.
::
sage: I = Ideal([])
Traceback (most recent call last):
...
ValueError: unable to determine which ring to embed the ideal in
sage: I = Ideal()
Traceback (most recent call last):
...
ValueError: need at least one argument
Note that some rings use different ideal implementations than the standard,
even if they are PIDs.::
sage: R.<x> = GF(5)[]
sage: I = R*(x^2+3)
sage: type(I)
<class 'sage.rings.polynomial.ideal.Ideal_1poly_field'>
You can also pass in a specific ideal type::
sage: from sage.rings.ideal import Ideal_pid
sage: I = Ideal(x^2+3,ideal_class=Ideal_pid)
sage: type(I)
<class 'sage.rings.ideal.Ideal_pid'>
TESTS::
sage: R, x = PolynomialRing(ZZ, 'x').objgen()
sage: I = R.ideal([4 + 3*x + x^2, 1 + x^2])
sage: I == loads(dumps(I))
True
::
sage: I = Ideal(R, [4 + 3*x + x^2, 1 + x^2])
sage: I == loads(dumps(I))
True
::
sage: I = Ideal((4 + 3*x + x^2, 1 + x^2))
sage: I == loads(dumps(I))
True
This shows that the issue at :trac:`5477` is fixed::
sage: R.<x> = QQ[]
sage: I = R.ideal([x + x^2])
sage: J = R.ideal([2*x + 2*x^2])
sage: J
Principal ideal (x^2 + x) of Univariate Polynomial Ring in x over Rational Field
sage: S = R.quotient_ring(I)
sage: U = R.quotient_ring(J)
sage: I == J
True
sage: S == U
True
"""
if len(args) == 0:
raise ValueError, "need at least one argument"
if len(args) == 1 and args[0] == []:
raise ValueError, "unable to determine which ring to embed the ideal in"
first = args[0]
if not isinstance(first, sage.rings.ring.Ring):
if isinstance(first, Ideal_generic) and len(args) == 1:
R = first.ring()
gens = first.gens()
else:
if isinstance(first, (list, tuple, GeneratorType)) and len(args) == 1:
gens = first
else:
gens = args
gens = Sequence(gens)
R = gens.universe()
else:
R = first
gens = args[1:]
if not commutative_ring.is_CommutativeRing(R):
raise TypeError, "R must be a commutative ring"
return R.ideal(*gens, **kwds)
def Ideal(*args, **kwds):
r"""
Create the ideal in ring with given generators.
There are some shorthand notations for creating an ideal, in
addition to using the :func:`Ideal` function:
- ``R.ideal(gens, coerce=True)``
- ``gens*R``
- ``R*gens``
INPUT:
- ``R`` - A ring (optional; if not given, will try to infer it from
``gens``)
- ``gens`` - list of elements generating the ideal
- ``coerce`` - bool (optional, default: ``True``);
whether ``gens`` need to be coerced into the ring.
OUTPUT: The ideal of ring generated by ``gens``.
EXAMPLES::
sage: R.<x> = ZZ[]
sage: I = R.ideal([4 + 3*x + x^2, 1 + x^2])
sage: I
Ideal (x^2 + 3*x + 4, x^2 + 1) of Univariate Polynomial Ring in x over Integer Ring
sage: Ideal(R, [4 + 3*x + x^2, 1 + x^2])
Ideal (x^2 + 3*x + 4, x^2 + 1) of Univariate Polynomial Ring in x over Integer Ring
sage: Ideal((4 + 3*x + x^2, 1 + x^2))
Ideal (x^2 + 3*x + 4, x^2 + 1) of Univariate Polynomial Ring in x over Integer Ring
::
sage: ideal(x^2-2*x+1, x^2-1)
Ideal (x^2 - 2*x + 1, x^2 - 1) of Univariate Polynomial Ring in x over Integer Ring
sage: ideal([x^2-2*x+1, x^2-1])
Ideal (x^2 - 2*x + 1, x^2 - 1) of Univariate Polynomial Ring in x over Integer Ring
sage: l = [x^2-2*x+1, x^2-1]
sage: ideal(f^2 for f in l)
Ideal (x^4 - 4*x^3 + 6*x^2 - 4*x + 1, x^4 - 2*x^2 + 1) of
Univariate Polynomial Ring in x over Integer Ring
This example illustrates how Sage finds a common ambient ring for
the ideal, even though 1 is in the integers (in this case).
::
sage: R.<t> = ZZ['t']
sage: i = ideal(1,t,t^2)
sage: i
Ideal (1, t, t^2) of Univariate Polynomial Ring in t over Integer Ring
sage: ideal(1/2,t,t^2)
Principal ideal (1) of Univariate Polynomial Ring in t over Rational Field
This shows that the issues at :trac:`1104` are resolved::
sage: Ideal(3, 5)
Principal ideal (1) of Integer Ring
sage: Ideal(ZZ, 3, 5)
Principal ideal (1) of Integer Ring
sage: Ideal(2, 4, 6)
Principal ideal (2) of Integer Ring
You have to provide enough information that Sage can figure out
which ring to put the ideal in.
::
sage: I = Ideal([])
Traceback (most recent call last):
...
ValueError: unable to determine which ring to embed the ideal in
sage: I = Ideal()
Traceback (most recent call last):
...
ValueError: need at least one argument
Note that some rings use different ideal implementations than the standard,
even if they are PIDs.::
sage: R.<x> = GF(5)[]
sage: I = R*(x^2+3)
sage: type(I)
<class 'sage.rings.polynomial.ideal.Ideal_1poly_field'>
You can also pass in a specific ideal type::
sage: from sage.rings.ideal import Ideal_pid
sage: I = Ideal(x^2+3,ideal_class=Ideal_pid)
sage: type(I)
<class 'sage.rings.ideal.Ideal_pid'>
TESTS::
sage: R.<x> = ZZ[]
sage: I = R.ideal([4 + 3*x + x^2, 1 + x^2])
sage: I == loads(dumps(I))
True
::
sage: I = Ideal(R, [4 + 3*x + x^2, 1 + x^2])
sage: I == loads(dumps(I))
True
::
sage: I = Ideal((4 + 3*x + x^2, 1 + x^2))
sage: I == loads(dumps(I))
True
This shows that the issue at :trac:`5477` is fixed::
sage: R.<x> = QQ[]
sage: I = R.ideal([x + x^2])
sage: J = R.ideal([2*x + 2*x^2])
sage: J
Principal ideal (x^2 + x) of Univariate Polynomial Ring in x over Rational Field
sage: S = R.quotient_ring(I)
sage: U = R.quotient_ring(J)
sage: I == J
True
sage: S == U
True
"""
if len(args) == 0:
raise ValueError("need at least one argument")
if len(args) == 1 and args[0] == []:
raise ValueError(
"unable to determine which ring to embed the ideal in")
first = args[0]
if not isinstance(first, sage.rings.ring.Ring):
if isinstance(first, Ideal_generic) and len(args) == 1:
R = first.ring()
gens = first.gens()
else:
if isinstance(first,
(list, tuple, GeneratorType)) and len(args) == 1:
gens = first
else:
gens = args
gens = Sequence(gens)
R = gens.universe()
else:
R = first
gens = args[1:]
if not commutative_ring.is_CommutativeRing(R):
raise TypeError("R must be a commutative ring")
return R.ideal(*gens, **kwds)