Sage provides functionality for computing with ideals. One can
create an ideal in any commutative ring by giving a
list of generators, using the notation
R.ideal([a,b,...]).
Ideal of cyclic n-roots from 1-st n variables of R if R is coercible to Singular. If n==None n is set to R.ngens()
INPUT:
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)
sage: I
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()
[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)
sage: B = I.groebner_basis()
sage: len(B)
45
Let q = R.base_ring().order() and (x0,...,x_n) = R.gens() then if q is finite this constructor returns
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
Another, similar example:
sage: Q.<x1,x2,x3,x4> = PolynomialRing(GF(2^4,name='alpha'), 4)
sage: J = sage.rings.ideal.FieldIdeal(Q); J
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
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:
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([])
...
ValueError: unable to determine which ring to embed the ideal in
sage: I = Ideal()
...
ValueError: need at least one argument
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
An ideal.
Return True if this ideal is not (0).
TESTS:
sage: I = ZZ.ideal(5)
sage: bool(I)
True
sage: I = ZZ['x'].ideal(0)
sage: bool(I)
False
sage: I = ZZ['x'].ideal(ZZ['x'].gen()^2)
sage: bool(I)
True
sage: I = QQ['x', 'y'].ideal(0)
sage: bool(I)
False
Apply the morphism phi to every element of this ideal. Returns an ideal in the domain of phi.
sage: psi = CC[‘x’].hom([-CC[‘x’].0]) sage: J = ideal([CC[‘x’].0 + 1]); J Principal ideal (1.00000000000000*x + 1.00000000000000) of Univariate Polynomial Ring in x over Complex Field with 53 bits of precision sage: psi(J) Principal ideal (-1.00000000000000*x + 1.00000000000000) of Univariate Polynomial Ring in x over Complex Field with 53 bits of precision sage: J.apply_morphism(psi) Principal ideal (-1.00000000000000*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
sage: K.<a> = NumberField(x^2 + 1) sage: A = K.ideal(a) sage: taus = K.embeddings(K) sage: A.apply_morphism(taus[0]) # identity Fractional ideal (a) sage: A.apply_morphism(taus[1]) # complex conjugation Fractional ideal (-a) sage: A.apply_morphism(taus[0]) == A.apply_morphism(taus[1]) True
sage: K.<a> = NumberField(x^2 + 5) sage: B = K.ideal([2, a + 1]); B Fractional ideal (2, a + 1) sage: taus = K.embeddings(K) sage: B.apply_morphism(taus[0]) # identity Fractional ideal (2, a + 1)
Since 2 is totally ramified, complex conjugation fixes it:
sage: B.apply_morphism(taus[1]) # complex conjugation Fractional ideal (2, a + 1)
sage: taus[1](B) Fractional ideal (2, a + 1)
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
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
And p-adic numbers:
sage: R = Zp(7, prec=10); R
7-adic Ring with capped relative precision 10
sage: I = 7*R; I
Principal ideal (7 + O(7^11)) of 7-adic Ring with capped relative precision 10
sage: I.base_ring()
7-adic Ring with capped relative precision 10
Return the category of this ideal.
EXAMPLES:
Note that category is dependent on the ring of the ideal.
sage: I = ZZ.ideal(7)
sage: J = ZZ[x].ideal(7,x)
sage: K = ZZ[x].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
Return a set of generators / a basis of self. This is usually the set of generators provided during object creation.
EXAMPLE:
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)
sage: ZZ.ideal(5,10).gens()
(5,)
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,)
Returns True if the ideal is maximal in the ring containing the ideal.
TODO: Make self.is_maximal() work! Write this code!
EXAMPLES:
sage: R = ZZ
sage: I = R.ideal(7)
sage: I.is_maximal()
...
NotImplementedError
Returns True if the ideal is prime in the ring containing the ideal.
TODO: Make self.is_prime() work! Write this code!
EXAMPLES:
sage: R = ZZ[x]
sage: I = R.ideal(7)
sage: I.is_prime()
...
NotImplementedError
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 intiialization of the ideal. (Can the base ring change? See example below.)
EXAMPLES:
sage: R = ZZ[x]
sage: I = R.ideal(2,x)
sage: I.is_principal()
...
NotImplementedError
sage: J = R.base_extend(QQ).ideal(2,x)
sage: J.is_principal()
True
Return True if this ideal is (0) or (1).
TESTS:
sage: I = ZZ.ideal(5)
sage: I.is_trivial()
False
sage: I = ZZ['x'].ideal(-1)
sage: I.is_trivial()
True
sage: I = ZZ['x'].ideal(ZZ['x'].gen()^2)
sage: I.is_trivial()
False
sage: I = QQ['x', 'y'].ideal(-5)
sage: I.is_trivial()
True
sage: I = CC['x'].ideal(0)
sage: I.is_trivial()
True
Return the reduction of the element of modulo the ideal
(=self). This is an element of
that is
equivalent modulo
to
.
EXAMPLES:
sage: ZZ.ideal(5).reduce(17)
2
sage: parent(ZZ.ideal(5).reduce(17))
Integer Ring
Returns 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
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
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')
sage: S = PolynomialRing(Rbar, 'y'); y = Rbar.gen(); S
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)
sage: J.ring()
Univariate Polynomial Ring in y over Univariate Quotient Polynomial Ring in a over Rational Field with modulus x^2 - 3
An ideal of a principal ideal domain.
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: 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
TESTS:
We cannot take the gcd of a principal ideal with a non-principal ideal as well: ( gcd(I,J) should be (7) )
sage: I = ZZ.ideal(7)
sage: J = ZZ[x].ideal(7,x)
sage: I.gcd(J)
...
NotImplementedError
sage: J.gcd(I)
...
AttributeError: 'Ideal_generic' object has no attribute 'gcd'
Note:
sage: type(I)
<class 'sage.rings.ideal.Ideal_pid'>
sage: type(J)
<class 'sage.rings.ideal.Ideal_generic'>
Returns 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()
True
Return the reduction of f modulo self.
EXAMPLES:
sage: I = 8*ZZ
sage: I.reduce(10)
2
sage: n = 10; n.mod(I)
2
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: 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)
...
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 Rational Field
sage: lift(F(12345/67890))
33
sage: (12345/67890) % 61
33
TESTS:
sage: ZZ.ideal(96).residue_field()
...
ValueError: The ideal (Principal ideal (96) of Integer Ring) is not prime
sage: R.<x>=QQ[]
sage: I=R.ideal(x^2+1)
sage: I.is_prime()
True
sage: I.residue_field()
Traceback (most recent call last):
NotImplementedError: residue_field() is only implemented for ZZ and rings of integers of number fields.
A principal ideal.
Returns True if x is in the ideal self.
EXAMPLES:
sage: P.<x> = PolynomialRing(ZZ)
sage: I = P.ideal(x^2-2)
sage: x^2 in I
False
sage: x^2-2 in I
True
sage: x^2-3 in I
False
Returns 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
Returns the generator of the principal ideal. The generators are elements 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
Note that the generator belongs to the ring from which the ideal was initialized:
sage: R = ZZ[x]
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
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 = ZZ[x]
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
n-th katsura ideal of R if R is coercible to Singular. If n==None n is set to R.ngens()
INPUT:
EXAMPLES:
sage: P.<x,y,z> = PolynomialRing(QQ,3)
sage: I = sage.rings.ideal.Katsura(P,3); I
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,1)
sage: J = sage.rings.ideal.Katsura(Q,1); J
Ideal (x - 1) of Multivariate Polynomial Ring in x over Rational Field
Returns 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
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