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)#
Ideal of cyclic
n
-roots from 1-stn
variables ofR
ifR
is coercible toSingular
.INPUT:
R
– base ring to construct ideal forn
– number of cyclic roots (default:None
). IfNone
, thenn
is set toR.ngens()
.homog
– (default:False
) ifTrue
a homogeneous ideal is returned using the last variable in the idealsingular
– singular instance to use
Note
R
will be set as the active ring inSingular
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
- sage.rings.ideal.FieldIdeal(R)#
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
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
- sage.rings.ideal.Ideal(*args, **kwds)#
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 fromgens
)gens
- list of elements generating the idealcoerce
- bool (optional, default:True
); whethergens
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 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'>
- class sage.rings.ideal.Ideal_fractional(ring, gens, coerce=True)#
Bases:
sage.rings.ideal.Ideal_generic
Fractional ideal of a ring.
See
Ideal()
.
- class sage.rings.ideal.Ideal_generic(ring, gens, coerce=True)#
Bases:
sage.structure.element.MonoidElement
An ideal.
See
Ideal()
.- absolute_norm()#
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_morphism(phi)#
Apply the morphism
phi
to every element of this ideal. Returns an ideal in the domain ofphi
.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
- associated_primes()#
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
- base_ring()#
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
- category()#
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
- embedded_primes()#
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]
- gen(i)#
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
- gens()#
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]
sage: ZZ.ideal(5,10).gens() (5,)
- gens_reduced()#
Same as
gens()
for this ideal, since there is currently no specialgens_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,)
- is_maximal()#
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
- is_primary(P=None)#
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() 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
Note
This uses the list of associated primes.
- is_prime()#
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.
- is_principal()#
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
- is_trivial()#
Return
True
if this ideal is \((0)\) or \((1)\).
- minimal_associated_primes()#
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
- ngens()#
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
- norm()#
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
- primary_decomposition()#
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
- random_element(*args, **kwds)#
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
- reduce(f)#
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
- ring()#
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
Note that
self.ring()
is different fromself.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
- class sage.rings.ideal.Ideal_pid(ring, gen)#
Bases:
sage.rings.ideal.Ideal_principal
An ideal of a principal ideal domain.
See
Ideal()
.- gcd(other)#
Returns the greatest common divisor of the principal ideal with the ideal
other
; that is, the largest principal ideal contained in both the ideal andother
Todo
This is not implemented in the case when
other
is neither principal nor when the generator ofself
is contained inother
. 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
- is_maximal()#
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
- is_prime()#
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
- reduce(f)#
Return the reduction of \(f\) modulo
self
.EXAMPLES:
sage: I = 8*ZZ sage: I.reduce(10) 2 sage: n = 10; n.mod(I) 2
- residue_field()#
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) 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
- class sage.rings.ideal.Ideal_principal(ring, gens, coerce=True)#
Bases:
sage.rings.ideal.Ideal_generic
A principal ideal.
See
Ideal()
.- divides(other)#
Return
True
ifself
dividesother
.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
- gen()#
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.<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
- is_principal()#
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
- sage.rings.ideal.Katsura(R, n=None, homog=False, singular=None)#
n
-th katsura ideal ofR
ifR
is coercible toSingular
.INPUT:
R
– base ring to construct ideal forn
– (default:None
) which katsura ideal ofR
. IfNone
, thenn
is set toR.ngens()
.homog
– ifTrue
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 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") sage: J = sage.rings.ideal.Katsura(Q,1); J Ideal (x - 1) of Multivariate Polynomial Ring in x over Rational Field
- sage.rings.ideal.is_Ideal(x)#
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