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.
Ideal of cyclic n-roots from 1-st n variables of R if R is coercible to Singular.
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 \((x_0,...,x_n)\) = 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:
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 ticket #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 ticket #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
Bases: sage.rings.ideal.Ideal_generic
Fractional ideal of a ring.
See Ideal().
Bases: sage.structure.element.MonoidElement
An ideal.
See Ideal().
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')[]
sage: I = R.ideal(t^4 + t + 1)
sage: I.absolute_norm()
Traceback (most recent call last):
...
NotImplementedError
Apply the morphism phi to every element of this ideal. Returns an ideal in the domain of phi.
EXAMPLES:
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
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
TESTS:
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)
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
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.
Note
category is dependent on the ring of the ideal.
EXAMPLES:
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 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()
[Ideal (y, x) of Multivariate Polynomial Ring in x, y over Rational Field]
Return the i-th generator in the current basis of this ideal.
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.gen(1)
y + 1
sage: ZZ.ideal(5,10).gen()
5
Return a set of generators / a basis of self.
This is the set of generators provided during creation of this ideal.
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,)
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
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 [AtiMac]). If this prime is \(P\), then \(I\) is said to be \(P\)-primary.
INPUT:
EXAMPLES:
sage: R.<x, y> = QQ[]
sage: I = R.ideal([x^2, x*y])
sage: I.is_primary()
False
sage: J = I.primary_decomposition()[1]; J
Ideal (y, x^2) of Multivariate Polynomial Ring in x, y over Rational Field
sage: J.is_primary()
True
sage: J.is_prime()
False
Some examples from the Macaulay2 documentation:
sage: R.<x, y, z> = GF(101)[]
sage: I = R.ideal([y^6])
sage: I.is_primary()
True
sage: I.is_primary(R.ideal([y]))
True
sage: I = R.ideal([x^4, y^7])
sage: I.is_primary()
True
sage: I = R.ideal([x*y, y^2])
sage: I.is_primary()
False
REFERENCES:
[AtiMac] | Atiyah and Macdonald, “Introduction to commutative algebra”, Addison-Wesley, 1969. |
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
True
sage: I = R.ideal([x^2-y])
sage: I.is_prime() # a non-maximal prime ideal
True
sage: I = R.ideal([x^2, y])
sage: I.is_prime() # a non-prime primary ideal
False
sage: I = R.ideal([x^2, x*y])
sage: I.is_prime() # a non-prime non-primary ideal
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
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
Note
For general rings, uses the list of associated primes.
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 = ZZ['x']
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
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 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
Return the number of generators in the basis.
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.ngens()
2
sage: ZZ.ideal(5,10).ngens()
1
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')[]
sage: I = R.ideal(t^4 + t + 1)
sage: I.norm()
Principal ideal (t^4 + t + 1) of Univariate Polynomial Ring in t over Finite Field in a of size 2^3
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
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
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
Bases: sage.rings.ideal.Ideal_principal
An ideal of a principal ideal domain.
See Ideal().
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
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)
Traceback (most recent call last):
...
NotImplementedError
sage: J.gcd(I)
Traceback (most recent call last):
...
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 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: 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
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()
True
In fields, only the zero ideal is prime:
sage: RR.ideal(0).is_prime()
True
sage: RR.ideal(7).is_prime()
False
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.
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)
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
TESTS:
sage: ZZ.ideal(96).residue_field()
Traceback (most recent call last):
...
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):
...
TypeError: residue fields only supported for polynomial rings over finite fields.
Bases: sage.rings.ideal.Ideal_generic
A principal ideal.
See Ideal().
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
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.
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
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
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