Ideals of commutative rings#

Sage provides functionality for computing with ideals. One can create an ideal in any commutative or non-commutative ring \(R\) by giving a list of generators, using the notation R.ideal([a,b,...]). The case of non-commutative rings is implemented in noncommutative_ideals.

A more convenient notation may be R*[a,b,...] or [a,b,...]*R. If \(R\) is non-commutative, the former creates a left and the latter a right ideal, and R*[a,b,...]*R creates a two-sided ideal.

sage.rings.ideal.Cyclic(R, n=None, homog=False, singular=None)[source]#

Ideal of cyclic n-roots from 1-st n variables of R if R is coercible to Singular.

INPUT:

  • R – base ring to construct ideal for

  • n – number of cyclic roots (default: None). If None, then n is set to R.ngens().

  • homog – (default: False) if True a homogeneous ideal is returned using the last variable in the ideal

  • singular – singular instance to use

Note

R will be set as the active ring in Singular

EXAMPLES:

An example from a multivariate polynomial ring over the rationals:

sage: P.<x,y,z> = PolynomialRing(QQ, 3, order='lex')
sage: I = sage.rings.ideal.Cyclic(P); I                                         # needs sage.libs.singular
Ideal (x + y + z, x*y + x*z + y*z, x*y*z - 1)
 of Multivariate Polynomial Ring in x, y, z over Rational Field
sage: I.groebner_basis()                                                        # needs sage.libs.singular
[x + y + z, y^2 + y*z + z^2, z^3 - 1]

>>> from sage.all import *
>>> P = PolynomialRing(QQ, Integer(3), order='lex', names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)
>>> I = sage.rings.ideal.Cyclic(P); I                                         # needs sage.libs.singular
Ideal (x + y + z, x*y + x*z + y*z, x*y*z - 1)
 of Multivariate Polynomial Ring in x, y, z over Rational Field
>>> I.groebner_basis()                                                        # needs sage.libs.singular
[x + y + z, y^2 + y*z + z^2, z^3 - 1]

We compute a Groebner basis for cyclic 6, which is a standard benchmark and test ideal:

sage: R.<x,y,z,t,u,v> = QQ['x,y,z,t,u,v']
sage: I = sage.rings.ideal.Cyclic(R, 6)                                         # needs sage.libs.singular
sage: B = I.groebner_basis()                                                    # needs sage.libs.singular
sage: len(B)                                                                    # needs sage.libs.singular
45

>>> from sage.all import *
>>> R = QQ['x,y,z,t,u,v']; (x, y, z, t, u, v,) = R._first_ngens(6)
>>> I = sage.rings.ideal.Cyclic(R, Integer(6))                                         # needs sage.libs.singular
>>> B = I.groebner_basis()                                                    # needs sage.libs.singular
>>> len(B)                                                                    # needs sage.libs.singular
45
sage.rings.ideal.FieldIdeal(R)[source]#

Let q = R.base_ring().order() and \((x_0,...,x_n)\) = R.gens() then if \(q\) is finite this constructor returns

\[\langle x_0^q - x_0, ... , x_n^q - x_n \rangle.\]

We call this ideal the field ideal and the generators the field equations.

EXAMPLES:

The field ideal generated from the polynomial ring over two variables in the finite field of size 2:

sage: P.<x,y> = PolynomialRing(GF(2), 2)
sage: I = sage.rings.ideal.FieldIdeal(P); I
Ideal (x^2 + x, y^2 + y) of
 Multivariate Polynomial Ring in x, y over Finite Field of size 2

>>> from sage.all import *
>>> P = PolynomialRing(GF(Integer(2)), Integer(2), names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> I = sage.rings.ideal.FieldIdeal(P); I
Ideal (x^2 + x, y^2 + y) of
 Multivariate Polynomial Ring in x, y over Finite Field of size 2

Another, similar example:

sage: Q.<x1,x2,x3,x4> = PolynomialRing(GF(2^4, name='alpha'), 4)                # needs sage.rings.finite_rings
sage: J = sage.rings.ideal.FieldIdeal(Q); J                                     # needs sage.rings.finite_rings
Ideal (x1^16 + x1, x2^16 + x2, x3^16 + x3, x4^16 + x4) of
 Multivariate Polynomial Ring in x1, x2, x3, x4
  over Finite Field in alpha of size 2^4

>>> from sage.all import *
>>> Q = PolynomialRing(GF(Integer(2)**Integer(4), name='alpha'), Integer(4), names=('x1', 'x2', 'x3', 'x4',)); (x1, x2, x3, x4,) = Q._first_ngens(4)# needs sage.rings.finite_rings
>>> J = sage.rings.ideal.FieldIdeal(Q); J                                     # needs sage.rings.finite_rings
Ideal (x1^16 + x1, x2^16 + x2, x3^16 + x3, x4^16 + x4) of
 Multivariate Polynomial Ring in x1, x2, x3, x4
  over Finite Field in alpha of size 2^4
sage.rings.ideal.Ideal(*args, **kwds)[source]#

Create the ideal in ring with given generators.

There are some shorthand notations for creating an ideal, in addition to using the 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 (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

>>> from sage.all import *
>>> R = ZZ['x']; (x,) = R._first_ngens(1)
>>> I = R.ideal([Integer(4) + Integer(3)*x + x**Integer(2), Integer(1) + x**Integer(2)])
>>> I
Ideal (x^2 + 3*x + 4, x^2 + 1) of Univariate Polynomial Ring in x over Integer Ring
>>> Ideal(R, [Integer(4) + Integer(3)*x + x**Integer(2), Integer(1) + x**Integer(2)])
Ideal (x^2 + 3*x + 4, x^2 + 1) of Univariate Polynomial Ring in x over Integer Ring
>>> Ideal((Integer(4) + Integer(3)*x + x**Integer(2), Integer(1) + x**Integer(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

>>> from sage.all import *
>>> ideal(x**Integer(2)-Integer(2)*x+Integer(1), x**Integer(2)-Integer(1))
Ideal (x^2 - 2*x + 1, x^2 - 1) of Univariate Polynomial Ring in x over Integer Ring
>>> ideal([x**Integer(2)-Integer(2)*x+Integer(1), x**Integer(2)-Integer(1)])
Ideal (x^2 - 2*x + 1, x^2 - 1) of Univariate Polynomial Ring in x over Integer Ring
>>> l = [x**Integer(2)-Integer(2)*x+Integer(1), x**Integer(2)-Integer(1)]
>>> ideal(f**Integer(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

>>> from sage.all import *
>>> R = ZZ['t']; (t,) = R._first_ngens(1)
>>> i = ideal(Integer(1),t,t**Integer(2))
>>> i
Ideal (1, t, t^2) of Univariate Polynomial Ring in t over Integer Ring
>>> ideal(Integer(1)/Integer(2),t,t**Integer(2))
Principal ideal (1) of Univariate Polynomial Ring in t over Rational Field

This shows that the issues at Issue #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

>>> from sage.all import *
>>> Ideal(Integer(3), Integer(5))
Principal ideal (1) of Integer Ring
>>> Ideal(ZZ, Integer(3), Integer(5))
Principal ideal (1) of Integer Ring
>>> Ideal(Integer(2), Integer(4), Integer(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

>>> from sage.all import *
>>> I = Ideal([])
Traceback (most recent call last):
...
ValueError: unable to determine which ring to embed the ideal in

>>> 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'>

>>> from sage.all import *
>>> R = GF(Integer(5))['x']; (x,) = R._first_ngens(1)
>>> I = R * (x**Integer(2) + Integer(3))
>>> 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'>

>>> from sage.all import *
>>> from sage.rings.ideal import Ideal_pid
>>> I = Ideal(x**Integer(2)+Integer(3),ideal_class=Ideal_pid)
>>> type(I)
<class 'sage.rings.ideal.Ideal_pid'>
class sage.rings.ideal.Ideal_fractional(ring, gens, coerce=True, **kwds)[source]#

Bases: Ideal_generic

Fractional ideal of a ring.

See Ideal().

class sage.rings.ideal.Ideal_generic(ring, gens, coerce=True, **kwds)[source]#

Bases: MonoidElement

An ideal.

See Ideal().

absolute_norm()[source]#

Returns the absolute norm of this ideal.

In the general case, this is just the ideal itself, since the ring it lies in can’t be implicitly assumed to be an extension of anything.

We include this function for compatibility with cases such as ideals in number fields.

Todo

Implement this method.

EXAMPLES:

sage: R.<t> = GF(9, names='a')[]                                            # needs sage.rings.finite_rings
sage: I = R.ideal(t^4 + t + 1)                                              # needs sage.rings.finite_rings
sage: I.absolute_norm()                                                     # needs sage.rings.finite_rings
Traceback (most recent call last):
...
NotImplementedError

>>> from sage.all import *
>>> R = GF(Integer(9), names='a')['t']; (t,) = R._first_ngens(1)# needs sage.rings.finite_rings
>>> I = R.ideal(t**Integer(4) + t + Integer(1))                                              # needs sage.rings.finite_rings
>>> I.absolute_norm()                                                     # needs sage.rings.finite_rings
Traceback (most recent call last):
...
NotImplementedError
apply_morphism(phi)[source]#

Apply the morphism phi to every element of this ideal. Returns an ideal in the domain of phi.

EXAMPLES:

sage: # needs sage.rings.real_mpfr
sage: psi = CC['x'].hom([-CC['x'].0])
sage: J = ideal([CC['x'].0 + 1]); J
Principal ideal (x + 1.00000000000000) of Univariate Polynomial Ring in x
 over Complex Field with 53 bits of precision
sage: psi(J)
Principal ideal (x - 1.00000000000000) of Univariate Polynomial Ring in x
 over Complex Field with 53 bits of precision
sage: J.apply_morphism(psi)
Principal ideal (x - 1.00000000000000) of Univariate Polynomial Ring in x
 over Complex Field with 53 bits of precision

>>> from sage.all import *
>>> # needs sage.rings.real_mpfr
>>> psi = CC['x'].hom([-CC['x'].gen(0)])
>>> J = ideal([CC['x'].gen(0) + Integer(1)]); J
Principal ideal (x + 1.00000000000000) of Univariate Polynomial Ring in x
 over Complex Field with 53 bits of precision
>>> psi(J)
Principal ideal (x - 1.00000000000000) of Univariate Polynomial Ring in x
 over Complex Field with 53 bits of precision
>>> J.apply_morphism(psi)
Principal ideal (x - 1.00000000000000) of Univariate Polynomial Ring in x
 over Complex Field with 53 bits of precision

sage: psi = ZZ['x'].hom([-ZZ['x'].0])
sage: J = ideal([ZZ['x'].0, 2]); J
Ideal (x, 2) of Univariate Polynomial Ring in x over Integer Ring
sage: psi(J)
Ideal (-x, 2) of Univariate Polynomial Ring in x over Integer Ring
sage: J.apply_morphism(psi)
Ideal (-x, 2) of Univariate Polynomial Ring in x over Integer Ring

>>> from sage.all import *
>>> psi = ZZ['x'].hom([-ZZ['x'].gen(0)])
>>> J = ideal([ZZ['x'].gen(0), Integer(2)]); J
Ideal (x, 2) of Univariate Polynomial Ring in x over Integer Ring
>>> psi(J)
Ideal (-x, 2) of Univariate Polynomial Ring in x over Integer Ring
>>> J.apply_morphism(psi)
Ideal (-x, 2) of Univariate Polynomial Ring in x over Integer Ring
associated_primes()[source]#

Return the list of associated prime ideals of this ideal.

EXAMPLES:

sage: R = ZZ['x']
sage: I = R.ideal(7)
sage: I.associated_primes()
Traceback (most recent call last):
...
NotImplementedError

>>> from sage.all import *
>>> R = ZZ['x']
>>> I = R.ideal(Integer(7))
>>> I.associated_primes()
Traceback (most recent call last):
...
NotImplementedError
base_ring()[source]#

Returns the base ring of this ideal.

EXAMPLES:

sage: R = ZZ
sage: I = 3*R; I
Principal ideal (3) of Integer Ring
sage: J = 2*I; J
Principal ideal (6) of Integer Ring
sage: I.base_ring(); J.base_ring()
Integer Ring
Integer Ring

>>> from sage.all import *
>>> R = ZZ
>>> I = Integer(3)*R; I
Principal ideal (3) of Integer Ring
>>> J = Integer(2)*I; J
Principal ideal (6) of Integer Ring
>>> I.base_ring(); J.base_ring()
Integer Ring
Integer Ring

We construct an example of an ideal of a quotient ring:

sage: R = PolynomialRing(QQ, 'x'); x = R.gen()
sage: I = R.ideal(x^2 - 2)
sage: I.base_ring()
Rational Field

>>> from sage.all import *
>>> R = PolynomialRing(QQ, 'x'); x = R.gen()
>>> I = R.ideal(x**Integer(2) - Integer(2))
>>> I.base_ring()
Rational Field

And \(p\)-adic numbers:

sage: R = Zp(7, prec=10); R                                                 # needs sage.rings.padics
7-adic Ring with capped relative precision 10
sage: I = 7*R; I                                                            # needs sage.rings.padics
Principal ideal (7 + O(7^11)) of 7-adic Ring with capped relative precision 10
sage: I.base_ring()                                                         # needs sage.rings.padics
7-adic Ring with capped relative precision 10

>>> from sage.all import *
>>> R = Zp(Integer(7), prec=Integer(10)); R                                                 # needs sage.rings.padics
7-adic Ring with capped relative precision 10
>>> I = Integer(7)*R; I                                                            # needs sage.rings.padics
Principal ideal (7 + O(7^11)) of 7-adic Ring with capped relative precision 10
>>> I.base_ring()                                                         # needs sage.rings.padics
7-adic Ring with capped relative precision 10
category()[source]#

Return the category of this ideal.

Note

category is dependent on the ring of the ideal.

EXAMPLES:

sage: P.<x> = ZZ[]
sage: I = ZZ.ideal(7)
sage: J = P.ideal(7,x)
sage: K = P.ideal(7)
sage: I.category()
Category of ring ideals in Integer Ring
sage: J.category()
Category of ring ideals in Univariate Polynomial Ring in x
over Integer Ring
sage: K.category()
Category of ring ideals in Univariate Polynomial Ring in x
over Integer Ring

>>> from sage.all import *
>>> P = ZZ['x']; (x,) = P._first_ngens(1)
>>> I = ZZ.ideal(Integer(7))
>>> J = P.ideal(Integer(7),x)
>>> K = P.ideal(Integer(7))
>>> I.category()
Category of ring ideals in Integer Ring
>>> J.category()
Category of ring ideals in Univariate Polynomial Ring in x
over Integer Ring
>>> K.category()
Category of ring ideals in Univariate Polynomial Ring in x
over Integer Ring
embedded_primes()[source]#

Return the list of embedded primes of this ideal.

EXAMPLES:

sage: R.<x, y> = QQ[]
sage: I = R.ideal(x^2, x*y)
sage: I.embedded_primes()                                                   # needs sage.libs.singular
[Ideal (y, x) of Multivariate Polynomial Ring in x, y over Rational Field]

>>> from sage.all import *
>>> R = QQ['x, y']; (x, y,) = R._first_ngens(2)
>>> I = R.ideal(x**Integer(2), x*y)
>>> I.embedded_primes()                                                   # needs sage.libs.singular
[Ideal (y, x) of Multivariate Polynomial Ring in x, y over Rational Field]
free_resolution(*args, **kwds)[source]#

Return a free resolution of self.

For input options, see FreeResolution.

EXAMPLES:

sage: R.<x> = PolynomialRing(QQ)
sage: I = R.ideal([x^4 + 3*x^2 + 2])
sage: I.free_resolution()                                                   # needs sage.modules
S^1 <-- S^1 <-- 0

>>> from sage.all import *
>>> R = PolynomialRing(QQ, names=('x',)); (x,) = R._first_ngens(1)
>>> I = R.ideal([x**Integer(4) + Integer(3)*x**Integer(2) + Integer(2)])
>>> I.free_resolution()                                                   # needs sage.modules
S^1 <-- S^1 <-- 0
gen(i)[source]#

Return the i-th generator in the current basis of this ideal.

EXAMPLES:

sage: P.<x,y> = PolynomialRing(QQ,2)
sage: I = Ideal([x,y+1]); I
Ideal (x, y + 1) of Multivariate Polynomial Ring in x, y over Rational Field
sage: I.gen(1)
y + 1

sage: ZZ.ideal(5,10).gen()
5

>>> from sage.all import *
>>> P = PolynomialRing(QQ,Integer(2), names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> I = Ideal([x,y+Integer(1)]); I
Ideal (x, y + 1) of Multivariate Polynomial Ring in x, y over Rational Field
>>> I.gen(Integer(1))
y + 1

>>> ZZ.ideal(Integer(5),Integer(10)).gen()
5
gens()[source]#

Return a set of generators / a basis of self.

This is the set of generators provided during creation of this ideal.

EXAMPLES:

sage: P.<x,y> = PolynomialRing(QQ,2)
sage: I = Ideal([x,y+1]); I
Ideal (x, y + 1) of Multivariate Polynomial Ring in x, y over Rational Field
sage: I.gens()
[x, y + 1]

>>> from sage.all import *
>>> P = PolynomialRing(QQ,Integer(2), names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> I = Ideal([x,y+Integer(1)]); I
Ideal (x, y + 1) of Multivariate Polynomial Ring in x, y over Rational Field
>>> I.gens()
[x, y + 1]

sage: ZZ.ideal(5,10).gens()
(5,)

>>> from sage.all import *
>>> ZZ.ideal(Integer(5),Integer(10)).gens()
(5,)
gens_reduced()[source]#

Same as gens() for this ideal, since there is currently no special gens_reduced algorithm implemented for this ring.

This method is provided so that ideals in \(\ZZ\) have the method gens_reduced(), just like ideals of number fields.

EXAMPLES:

sage: ZZ.ideal(5).gens_reduced()
(5,)

>>> from sage.all import *
>>> ZZ.ideal(Integer(5)).gens_reduced()
(5,)
graded_free_resolution(*args, **kwds)[source]#

Return a graded free resolution of self.

For input options, see GradedFiniteFreeResolution.

EXAMPLES:

sage: R.<x> = PolynomialRing(QQ)
sage: I = R.ideal([x^3])
sage: I.graded_free_resolution()                                            # needs sage.modules
S(0) <-- S(-3) <-- 0

>>> from sage.all import *
>>> R = PolynomialRing(QQ, names=('x',)); (x,) = R._first_ngens(1)
>>> I = R.ideal([x**Integer(3)])
>>> I.graded_free_resolution()                                            # needs sage.modules
S(0) <-- S(-3) <-- 0
is_maximal()[source]#

Return True if the ideal is maximal in the ring containing the ideal.

Todo

This is not implemented for many rings. Implement it!

EXAMPLES:

sage: R = ZZ
sage: I = R.ideal(7)
sage: I.is_maximal()
True
sage: R.ideal(16).is_maximal()
False
sage: S = Integers(8)
sage: S.ideal(0).is_maximal()
False
sage: S.ideal(2).is_maximal()
True
sage: S.ideal(4).is_maximal()
False

>>> from sage.all import *
>>> R = ZZ
>>> I = R.ideal(Integer(7))
>>> I.is_maximal()
True
>>> R.ideal(Integer(16)).is_maximal()
False
>>> S = Integers(Integer(8))
>>> S.ideal(Integer(0)).is_maximal()
False
>>> S.ideal(Integer(2)).is_maximal()
True
>>> S.ideal(Integer(4)).is_maximal()
False
is_primary(P=None)[source]#

Returns True if this ideal is primary (or \(P\)-primary, if a prime ideal \(P\) is specified).

Recall that an ideal \(I\) is primary if and only if \(I\) has a unique associated prime (see page 52 in [AM1969]). If this prime is \(P\), then \(I\) is said to be \(P\)-primary.

INPUT:

  • P – (default: None) a prime ideal in the same ring

EXAMPLES:

sage: R.<x, y> = QQ[]
sage: I = R.ideal([x^2, x*y])
sage: I.is_primary()                                                        # needs sage.libs.singular
False
sage: J = I.primary_decomposition()[1]; J                                   # needs sage.libs.singular
Ideal (y, x^2) of Multivariate Polynomial Ring in x, y over Rational Field
sage: J.is_primary()                                                        # needs sage.libs.singular
True
sage: J.is_prime()                                                          # needs sage.libs.singular
False

>>> from sage.all import *
>>> R = QQ['x, y']; (x, y,) = R._first_ngens(2)
>>> I = R.ideal([x**Integer(2), x*y])
>>> I.is_primary()                                                        # needs sage.libs.singular
False
>>> J = I.primary_decomposition()[Integer(1)]; J                                   # needs sage.libs.singular
Ideal (y, x^2) of Multivariate Polynomial Ring in x, y over Rational Field
>>> J.is_primary()                                                        # needs sage.libs.singular
True
>>> J.is_prime()                                                          # needs sage.libs.singular
False

Some examples from the Macaulay2 documentation:

sage: # needs sage.rings.finite_rings
sage: R.<x, y, z> = GF(101)[]
sage: I = R.ideal([y^6])
sage: I.is_primary()                                                        # needs sage.libs.singular
True
sage: I.is_primary(R.ideal([y]))                                            # needs sage.libs.singular
True
sage: I = R.ideal([x^4, y^7])
sage: I.is_primary()                                                        # needs sage.libs.singular
True
sage: I = R.ideal([x*y, y^2])
sage: I.is_primary()                                                        # needs sage.libs.singular
False

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> R = GF(Integer(101))['x, y, z']; (x, y, z,) = R._first_ngens(3)
>>> I = R.ideal([y**Integer(6)])
>>> I.is_primary()                                                        # needs sage.libs.singular
True
>>> I.is_primary(R.ideal([y]))                                            # needs sage.libs.singular
True
>>> I = R.ideal([x**Integer(4), y**Integer(7)])
>>> I.is_primary()                                                        # needs sage.libs.singular
True
>>> I = R.ideal([x*y, y**Integer(2)])
>>> I.is_primary()                                                        # needs sage.libs.singular
False

Note

This uses the list of associated primes.

is_prime()[source]#

Return True if this ideal is prime.

EXAMPLES:

sage: R.<x, y> = QQ[]
sage: I = R.ideal([x, y])
sage: I.is_prime()        # a maximal ideal                                 # needs sage.libs.singular
True
sage: I = R.ideal([x^2 - y])
sage: I.is_prime()        # a non-maximal prime ideal                       # needs sage.libs.singular
True
sage: I = R.ideal([x^2, y])
sage: I.is_prime()        # a non-prime primary ideal                       # needs sage.libs.singular
False
sage: I = R.ideal([x^2, x*y])
sage: I.is_prime()        # a non-prime non-primary ideal                   # needs sage.libs.singular
False

sage: S = Integers(8)
sage: S.ideal(0).is_prime()
False
sage: S.ideal(2).is_prime()
True
sage: S.ideal(4).is_prime()
False

>>> from sage.all import *
>>> R = QQ['x, y']; (x, y,) = R._first_ngens(2)
>>> I = R.ideal([x, y])
>>> I.is_prime()        # a maximal ideal                                 # needs sage.libs.singular
True
>>> I = R.ideal([x**Integer(2) - y])
>>> I.is_prime()        # a non-maximal prime ideal                       # needs sage.libs.singular
True
>>> I = R.ideal([x**Integer(2), y])
>>> I.is_prime()        # a non-prime primary ideal                       # needs sage.libs.singular
False
>>> I = R.ideal([x**Integer(2), x*y])
>>> I.is_prime()        # a non-prime non-primary ideal                   # needs sage.libs.singular
False

>>> S = Integers(Integer(8))
>>> S.ideal(Integer(0)).is_prime()
False
>>> S.ideal(Integer(2)).is_prime()
True
>>> S.ideal(Integer(4)).is_prime()
False

Note that this method is not implemented for all rings where it could be:

sage: R.<x> = ZZ[]
sage: I = R.ideal(7)
sage: I.is_prime()        # when implemented, should be True
Traceback (most recent call last):
...
NotImplementedError

>>> from sage.all import *
>>> R = ZZ['x']; (x,) = R._first_ngens(1)
>>> I = R.ideal(Integer(7))
>>> I.is_prime()        # when implemented, should be True
Traceback (most recent call last):
...
NotImplementedError

Note

For general rings, uses the list of associated primes.

is_principal()[source]#

Returns True if the ideal is principal in the ring containing the ideal.

Todo

Code is naive. Only keeps track of ideal generators as set during initialization of the ideal. (Can the base ring change? See example below.)

EXAMPLES:

sage: R.<x> = ZZ[]
sage: I = R.ideal(2, x)
sage: I.is_principal()
Traceback (most recent call last):
...
NotImplementedError
sage: J = R.base_extend(QQ).ideal(2, x)
sage: J.is_principal()
True

>>> from sage.all import *
>>> R = ZZ['x']; (x,) = R._first_ngens(1)
>>> I = R.ideal(Integer(2), x)
>>> I.is_principal()
Traceback (most recent call last):
...
NotImplementedError
>>> J = R.base_extend(QQ).ideal(Integer(2), x)
>>> J.is_principal()
True
is_trivial()[source]#

Return True if this ideal is \((0)\) or \((1)\).

minimal_associated_primes()[source]#

Return the list of minimal associated prime ideals of this ideal.

EXAMPLES:

sage: R = ZZ['x']
sage: I = R.ideal(7)
sage: I.minimal_associated_primes()
Traceback (most recent call last):
...
NotImplementedError

>>> from sage.all import *
>>> R = ZZ['x']
>>> I = R.ideal(Integer(7))
>>> I.minimal_associated_primes()
Traceback (most recent call last):
...
NotImplementedError
ngens()[source]#

Return the number of generators in the basis.

EXAMPLES:

sage: P.<x,y> = PolynomialRing(QQ,2)
sage: I = Ideal([x,y+1]); I
Ideal (x, y + 1) of Multivariate Polynomial Ring in x, y over Rational Field
sage: I.ngens()
2

sage: ZZ.ideal(5,10).ngens()
1

>>> from sage.all import *
>>> P = PolynomialRing(QQ,Integer(2), names=('x', 'y',)); (x, y,) = P._first_ngens(2)
>>> I = Ideal([x,y+Integer(1)]); I
Ideal (x, y + 1) of Multivariate Polynomial Ring in x, y over Rational Field
>>> I.ngens()
2

>>> ZZ.ideal(Integer(5),Integer(10)).ngens()
1
norm()[source]#

Returns the norm of this ideal.

In the general case, this is just the ideal itself, since the ring it lies in can’t be implicitly assumed to be an extension of anything.

We include this function for compatibility with cases such as ideals in number fields.

EXAMPLES:

sage: R.<t> = GF(8, names='a')[]                                            # needs sage.rings.finite_rings
sage: I = R.ideal(t^4 + t + 1)                                              # needs sage.rings.finite_rings
sage: I.norm()                                                              # needs sage.rings.finite_rings
Principal ideal (t^4 + t + 1) of Univariate Polynomial Ring in t
 over Finite Field in a of size 2^3

>>> from sage.all import *
>>> R = GF(Integer(8), names='a')['t']; (t,) = R._first_ngens(1)# needs sage.rings.finite_rings
>>> I = R.ideal(t**Integer(4) + t + Integer(1))                                              # needs sage.rings.finite_rings
>>> I.norm()                                                              # needs sage.rings.finite_rings
Principal ideal (t^4 + t + 1) of Univariate Polynomial Ring in t
 over Finite Field in a of size 2^3
primary_decomposition()[source]#

Return a decomposition of this ideal into primary ideals.

EXAMPLES:

sage: R = ZZ['x']
sage: I = R.ideal(7)
sage: I.primary_decomposition()
Traceback (most recent call last):
...
NotImplementedError

>>> from sage.all import *
>>> R = ZZ['x']
>>> I = R.ideal(Integer(7))
>>> I.primary_decomposition()
Traceback (most recent call last):
...
NotImplementedError
random_element(*args, **kwds)[source]#

Return a random element in this ideal.

EXAMPLES:

sage: P.<a,b,c> = GF(5)[[]]
sage: I = P.ideal([a^2, a*b + c, c^3])
sage: I.random_element()  # random
2*a^5*c + a^2*b*c^4 + ... + O(a, b, c)^13

>>> from sage.all import *
>>> P = GF(Integer(5))[['a, b, c']]; (a, b, c,) = P._first_ngens(3)
>>> I = P.ideal([a**Integer(2), a*b + c, c**Integer(3)])
>>> I.random_element()  # random
2*a^5*c + a^2*b*c^4 + ... + O(a, b, c)^13
reduce(f)[source]#

Return the reduction of the element of \(f\) modulo self.

This is an element of \(R\) that is equivalent modulo \(I\) to \(f\) where \(I\) is self.

EXAMPLES:

sage: ZZ.ideal(5).reduce(17)
2
sage: parent(ZZ.ideal(5).reduce(17))
Integer Ring

>>> from sage.all import *
>>> ZZ.ideal(Integer(5)).reduce(Integer(17))
2
>>> parent(ZZ.ideal(Integer(5)).reduce(Integer(17)))
Integer Ring
ring()[source]#

Return the ring containing this ideal.

EXAMPLES:

sage: R = ZZ
sage: I = 3*R; I
Principal ideal (3) of Integer Ring
sage: J = 2*I; J
Principal ideal (6) of Integer Ring
sage: I.ring(); J.ring()
Integer Ring
Integer Ring

>>> from sage.all import *
>>> R = ZZ
>>> I = Integer(3)*R; I
Principal ideal (3) of Integer Ring
>>> J = Integer(2)*I; J
Principal ideal (6) of Integer Ring
>>> I.ring(); J.ring()
Integer Ring
Integer Ring

Note that self.ring() is different from self.base_ring()

sage: R = PolynomialRing(QQ, 'x'); x = R.gen()
sage: I = R.ideal(x^2 - 2)
sage: I.base_ring()
Rational Field
sage: I.ring()
Univariate Polynomial Ring in x over Rational Field

>>> from sage.all import *
>>> R = PolynomialRing(QQ, 'x'); x = R.gen()
>>> I = R.ideal(x**Integer(2) - Integer(2))
>>> I.base_ring()
Rational Field
>>> I.ring()
Univariate Polynomial Ring in x over Rational Field

Another example using polynomial rings:

sage: R = PolynomialRing(QQ, 'x'); x = R.gen()
sage: I = R.ideal(x^2 - 3)
sage: I.ring()
Univariate Polynomial Ring in x over Rational Field
sage: Rbar = R.quotient(I, names='a')                                       # needs sage.libs.pari
sage: S = PolynomialRing(Rbar, 'y'); y = Rbar.gen(); S                      # needs sage.libs.pari
Univariate Polynomial Ring in y over
 Univariate Quotient Polynomial Ring in a over Rational Field with modulus x^2 - 3
sage: J = S.ideal(y^2 + 1)                                                  # needs sage.libs.pari
sage: J.ring()                                                              # needs sage.libs.pari
Univariate Polynomial Ring in y over
 Univariate Quotient Polynomial Ring in a over Rational Field with modulus x^2 - 3

>>> from sage.all import *
>>> R = PolynomialRing(QQ, 'x'); x = R.gen()
>>> I = R.ideal(x**Integer(2) - Integer(3))
>>> I.ring()
Univariate Polynomial Ring in x over Rational Field
>>> Rbar = R.quotient(I, names='a')                                       # needs sage.libs.pari
>>> S = PolynomialRing(Rbar, 'y'); y = Rbar.gen(); S                      # needs sage.libs.pari
Univariate Polynomial Ring in y over
 Univariate Quotient Polynomial Ring in a over Rational Field with modulus x^2 - 3
>>> J = S.ideal(y**Integer(2) + Integer(1))                                                  # needs sage.libs.pari
>>> J.ring()                                                              # needs sage.libs.pari
Univariate Polynomial Ring in y over
 Univariate Quotient Polynomial Ring in a over Rational Field with modulus x^2 - 3
class sage.rings.ideal.Ideal_pid(ring, gens, coerce=True, **kwds)[source]#

Bases: Ideal_principal

An ideal of a principal ideal domain.

See Ideal().

EXAMPLES:

sage: I = 8*ZZ
sage: I
Principal ideal (8) of Integer Ring

>>> from sage.all import *
>>> I = Integer(8)*ZZ
>>> I
Principal ideal (8) of Integer Ring
gcd(other)[source]#

Returns the greatest common divisor of the principal ideal with the ideal other; that is, the largest principal ideal contained in both the ideal and other

Todo

This is not implemented in the case when other is neither principal nor when the generator of self is contained in other. Also, it seems that this class is used only in PIDs–is this redundant?

Note

The second example is broken.

EXAMPLES:

An example in the principal ideal domain \(\ZZ\):

sage: R = ZZ
sage: I = R.ideal(42)
sage: J = R.ideal(70)
sage: I.gcd(J)
Principal ideal (14) of Integer Ring
sage: J.gcd(I)
Principal ideal (14) of Integer Ring

>>> from sage.all import *
>>> R = ZZ
>>> I = R.ideal(Integer(42))
>>> J = R.ideal(Integer(70))
>>> I.gcd(J)
Principal ideal (14) of Integer Ring
>>> J.gcd(I)
Principal ideal (14) of Integer Ring
is_maximal()[source]#

Returns whether this ideal is maximal.

Principal ideal domains have Krull dimension 1 (or 0), so an ideal is maximal if and only if it’s prime (and nonzero if the ring is not a field).

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: R.<t> = GF(5)[]
sage: p = R.ideal(t^2 + 2)
sage: p.is_maximal()
True
sage: p = R.ideal(t^2 + 1)
sage: p.is_maximal()
False
sage: p = R.ideal(0)
sage: p.is_maximal()
False
sage: p = R.ideal(1)
sage: p.is_maximal()
False

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> R = GF(Integer(5))['t']; (t,) = R._first_ngens(1)
>>> p = R.ideal(t**Integer(2) + Integer(2))
>>> p.is_maximal()
True
>>> p = R.ideal(t**Integer(2) + Integer(1))
>>> p.is_maximal()
False
>>> p = R.ideal(Integer(0))
>>> p.is_maximal()
False
>>> p = R.ideal(Integer(1))
>>> p.is_maximal()
False
is_prime()[source]#

Return True if the ideal is prime.

This relies on the ring elements having a method is_irreducible() implemented, since an ideal \((a)\) is prime iff \(a\) is irreducible (or 0).

EXAMPLES:

sage: ZZ.ideal(2).is_prime()
True
sage: ZZ.ideal(-2).is_prime()
True
sage: ZZ.ideal(4).is_prime()
False
sage: ZZ.ideal(0).is_prime()
True
sage: R.<x> = QQ[]
sage: P = R.ideal(x^2 + 1); P
Principal ideal (x^2 + 1) of Univariate Polynomial Ring in x over Rational Field
sage: P.is_prime()                                                          # needs sage.libs.pari
True

>>> from sage.all import *
>>> ZZ.ideal(Integer(2)).is_prime()
True
>>> ZZ.ideal(-Integer(2)).is_prime()
True
>>> ZZ.ideal(Integer(4)).is_prime()
False
>>> ZZ.ideal(Integer(0)).is_prime()
True
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> P = R.ideal(x**Integer(2) + Integer(1)); P
Principal ideal (x^2 + 1) of Univariate Polynomial Ring in x over Rational Field
>>> P.is_prime()                                                          # needs sage.libs.pari
True

In fields, only the zero ideal is prime:

sage: RR.ideal(0).is_prime()
True
sage: RR.ideal(7).is_prime()
False

>>> from sage.all import *
>>> RR.ideal(Integer(0)).is_prime()
True
>>> RR.ideal(Integer(7)).is_prime()
False
reduce(f)[source]#

Return the reduction of \(f\) modulo self.

EXAMPLES:

sage: I = 8*ZZ
sage: I.reduce(10)
2
sage: n = 10; n.mod(I)
2

>>> from sage.all import *
>>> I = Integer(8)*ZZ
>>> I.reduce(Integer(10))
2
>>> n = Integer(10); n.mod(I)
2
residue_field()[source]#

Return the residue class field of this ideal, which must be prime.

Todo

Implement this for more general rings. Currently only defined for \(\ZZ\) and for number field orders.

EXAMPLES:

sage: # needs sage.libs.pari
sage: P = ZZ.ideal(61); P
Principal ideal (61) of Integer Ring
sage: F = P.residue_field(); F
Residue field of Integers modulo 61
sage: pi = F.reduction_map(); pi
Partially defined reduction map:
  From: Rational Field
  To:   Residue field of Integers modulo 61
sage: pi(123/234)
6
sage: pi(1/61)
Traceback (most recent call last):
...
ZeroDivisionError: Cannot reduce rational 1/61 modulo 61: it has negative valuation
sage: lift = F.lift_map(); lift
Lifting map:
  From: Residue field of Integers modulo 61
  To:   Integer Ring
sage: lift(F(12345/67890))
33
sage: (12345/67890) % 61
33

>>> from sage.all import *
>>> # needs sage.libs.pari
>>> P = ZZ.ideal(Integer(61)); P
Principal ideal (61) of Integer Ring
>>> F = P.residue_field(); F
Residue field of Integers modulo 61
>>> pi = F.reduction_map(); pi
Partially defined reduction map:
  From: Rational Field
  To:   Residue field of Integers modulo 61
>>> pi(Integer(123)/Integer(234))
6
>>> pi(Integer(1)/Integer(61))
Traceback (most recent call last):
...
ZeroDivisionError: Cannot reduce rational 1/61 modulo 61: it has negative valuation
>>> lift = F.lift_map(); lift
Lifting map:
  From: Residue field of Integers modulo 61
  To:   Integer Ring
>>> lift(F(Integer(12345)/Integer(67890)))
33
>>> (Integer(12345)/Integer(67890)) % Integer(61)
33
class sage.rings.ideal.Ideal_principal(ring, gens, coerce=True, **kwds)[source]#

Bases: Ideal_generic

A principal ideal.

See Ideal().

divides(other)[source]#

Return True if self divides other.

EXAMPLES:

sage: P.<x> = PolynomialRing(QQ)
sage: I = P.ideal(x)
sage: J = P.ideal(x^2)
sage: I.divides(J)
True
sage: J.divides(I)
False

>>> from sage.all import *
>>> P = PolynomialRing(QQ, names=('x',)); (x,) = P._first_ngens(1)
>>> I = P.ideal(x)
>>> J = P.ideal(x**Integer(2))
>>> I.divides(J)
True
>>> J.divides(I)
False
gen(i=0)[source]#

Return the generator of the principal ideal.

The generator is an element of the ring containing the ideal.

EXAMPLES:

A simple example in the integers:

sage: R = ZZ
sage: I = R.ideal(7)
sage: J = R.ideal(7, 14)
sage: I.gen(); J.gen()
7
7

>>> from sage.all import *
>>> R = ZZ
>>> I = R.ideal(Integer(7))
>>> J = R.ideal(Integer(7), Integer(14))
>>> I.gen(); J.gen()
7
7

Note that the generator belongs to the ring from which the ideal was initialized:

sage: R.<x> = ZZ[]
sage: I = R.ideal(x)
sage: J = R.base_extend(QQ).ideal(2,x)
sage: a = I.gen(); a
x
sage: b = J.gen(); b
1
sage: a.base_ring()
Integer Ring
sage: b.base_ring()
Rational Field

>>> from sage.all import *
>>> R = ZZ['x']; (x,) = R._first_ngens(1)
>>> I = R.ideal(x)
>>> J = R.base_extend(QQ).ideal(Integer(2),x)
>>> a = I.gen(); a
x
>>> b = J.gen(); b
1
>>> a.base_ring()
Integer Ring
>>> b.base_ring()
Rational Field
is_principal()[source]#

Returns True if the ideal is principal in the ring containing the ideal. When the ideal construction is explicitly principal (i.e. when we define an ideal with one element) this is always the case.

EXAMPLES:

Note that Sage automatically coerces ideals into principal ideals during initialization:

sage: R.<x> = ZZ[]
sage: I = R.ideal(x)
sage: J = R.ideal(2,x)
sage: K = R.base_extend(QQ).ideal(2,x)
sage: I
Principal ideal (x) of Univariate Polynomial Ring in x
over Integer Ring
sage: J
Ideal (2, x) of Univariate Polynomial Ring in x over Integer Ring
sage: K
Principal ideal (1) of Univariate Polynomial Ring in x
over Rational Field
sage: I.is_principal()
True
sage: K.is_principal()
True

>>> from sage.all import *
>>> R = ZZ['x']; (x,) = R._first_ngens(1)
>>> I = R.ideal(x)
>>> J = R.ideal(Integer(2),x)
>>> K = R.base_extend(QQ).ideal(Integer(2),x)
>>> I
Principal ideal (x) of Univariate Polynomial Ring in x
over Integer Ring
>>> J
Ideal (2, x) of Univariate Polynomial Ring in x over Integer Ring
>>> K
Principal ideal (1) of Univariate Polynomial Ring in x
over Rational Field
>>> I.is_principal()
True
>>> K.is_principal()
True
sage.rings.ideal.Katsura(R, n=None, homog=False, singular=None)[source]#

n-th katsura ideal of R if R is coercible to Singular.

INPUT:

  • R – base ring to construct ideal for

  • n – (default: None) which katsura ideal of R. If None, then n is set to R.ngens().

  • homog – if True a homogeneous ideal is returned using the last variable in the ideal (default: False)

  • singular – singular instance to use

EXAMPLES:

sage: P.<x,y,z> = PolynomialRing(QQ, 3)
sage: I = sage.rings.ideal.Katsura(P, 3); I                                     # needs sage.libs.singular
Ideal (x + 2*y + 2*z - 1, x^2 + 2*y^2 + 2*z^2 - x, 2*x*y + 2*y*z - y)
of Multivariate Polynomial Ring in x, y, z over Rational Field

>>> from sage.all import *
>>> P = PolynomialRing(QQ, Integer(3), names=('x', 'y', 'z',)); (x, y, z,) = P._first_ngens(3)
>>> I = sage.rings.ideal.Katsura(P, Integer(3)); I                                     # needs sage.libs.singular
Ideal (x + 2*y + 2*z - 1, x^2 + 2*y^2 + 2*z^2 - x, 2*x*y + 2*y*z - y)
of Multivariate Polynomial Ring in x, y, z over Rational Field

sage: Q.<x> = PolynomialRing(QQ, implementation="singular")                     # needs sage.libs.singular
sage: J = sage.rings.ideal.Katsura(Q,1); J                                      # needs sage.libs.singular
Ideal (x - 1) of Multivariate Polynomial Ring in x over Rational Field

>>> from sage.all import *
>>> Q = PolynomialRing(QQ, implementation="singular", names=('x',)); (x,) = Q._first_ngens(1)# needs sage.libs.singular
>>> J = sage.rings.ideal.Katsura(Q,Integer(1)); J                                      # needs sage.libs.singular
Ideal (x - 1) of Multivariate Polynomial Ring in x over Rational Field
sage.rings.ideal.is_Ideal(x)[source]#

Return True if object is an ideal of a ring.

EXAMPLES:

A simple example involving the ring of integers. Note that Sage does not interpret rings objects themselves as ideals. However, one can still explicitly construct these ideals:

sage: from sage.rings.ideal import is_Ideal
sage: R = ZZ
sage: is_Ideal(R)
False
sage: 1*R; is_Ideal(1*R)
Principal ideal (1) of Integer Ring
True
sage: 0*R; is_Ideal(0*R)
Principal ideal (0) of Integer Ring
True

>>> from sage.all import *
>>> from sage.rings.ideal import is_Ideal
>>> R = ZZ
>>> is_Ideal(R)
False
>>> Integer(1)*R; is_Ideal(Integer(1)*R)
Principal ideal (1) of Integer Ring
True
>>> Integer(0)*R; is_Ideal(Integer(0)*R)
Principal ideal (0) of Integer Ring
True

Sage recognizes ideals of polynomial rings as well:

sage: R = PolynomialRing(QQ, 'x'); x = R.gen()
sage: I = R.ideal(x^2 + 1); I
Principal ideal (x^2 + 1) of Univariate Polynomial Ring in x over Rational Field
sage: is_Ideal(I)
True
sage: is_Ideal((x^2 + 1)*R)
True

>>> from sage.all import *
>>> R = PolynomialRing(QQ, 'x'); x = R.gen()
>>> I = R.ideal(x**Integer(2) + Integer(1)); I
Principal ideal (x^2 + 1) of Univariate Polynomial Ring in x over Rational Field
>>> is_Ideal(I)
True
>>> is_Ideal((x**Integer(2) + Integer(1))*R)
True