Rings

This module provides the abstract base class Ring from which all rings in Sage (used to) derive, as well as a selection of more specific base classes.

Warning

Those classes, except maybe for the lowest ones like Ring, CommutativeRing, Algebra and CommutativeAlgebra, are being progressively deprecated in favor of the corresponding categories. which are more flexible, in particular with respect to multiple inheritance.

The class inheritance hierarchy is:

• Ring

  • Algebra

  • CommutativeRing

    • NoetherianRing

    • CommutativeAlgebra

    • IntegralDomain

      • DedekindDomain

      • PrincipalIdealDomain

Subclasses of PrincipalIdealDomain are

• EuclideanDomain

• Field

  • FiniteField

Some aspects of this structure may seem strange, but this is an unfortunate consequence of the fact that Cython classes do not support multiple inheritance. Hence, for instance, Field cannot be a subclass of both NoetherianRing and PrincipalIdealDomain, although all fields are Noetherian PIDs.

(A distinct but equally awkward issue is that sometimes we may not know in advance whether or not a ring belongs in one of these classes; e.g. some orders in number fields are Dedekind domains, but others are not, and we still want to offer a unified interface, so orders are never instances of the DedekindDomain class.)

AUTHORS:

• David Harvey (2006-10-16): changed CommutativeAlgebra to derive from CommutativeRing instead of from Algebra.

• David Loeffler (2009-07-09): documentation fixes, added to reference manual.

• Simon King (2011-03-29): Proper use of the category framework for rings.

• Simon King (2011-05-20): Modify multiplication and _ideal_class_ to support ideals of non-commutative rings.

class sage.rings.ring.Algebra

Bases: sage.rings.ring.Ring

Generic algebra

characteristic()

Return the characteristic of this algebra, which is the same as the characteristic of its base ring.

See objects with the base_ring attribute for additional examples. Here are some examples that explicitly use the Algebra class.

EXAMPLES:

sage: A = Algebra(ZZ); A
<sage.rings.ring.Algebra object at ...>
sage: A.characteristic()
0
sage: A = Algebra(GF(7^3, 'a'))
sage: A.characteristic()
7

has_standard_involution()

Return True if the algebra has a standard involution and False otherwise. This algorithm follows Algorithm 2.10 from John Voight’s Identifying the Matrix Ring. Currently the only type of algebra this will work for is a quaternion algebra. Though this function seems redundant, once algebras have more functionality, in particular have a method to construct a basis, this algorithm will have more general purpose.

EXAMPLES:

sage: B = QuaternionAlgebra(2)
sage: B.has_standard_involution()
True
sage: R.<x> = PolynomialRing(QQ)
sage: K.<u> = NumberField(x**2 - 2)
sage: A = QuaternionAlgebra(K,-2,5)
sage: A.has_standard_involution()
True
sage: L.<a,b> = FreeAlgebra(QQ,2)
sage: L.has_standard_involution()
Traceback (most recent call last):
...
NotImplementedError: has_standard_involution is not implemented for this algebra

class sage.rings.ring.CommutativeAlgebra

Bases: sage.rings.ring.CommutativeRing

Generic commutative algebra

is_commutative()

Return True since this algebra is commutative.

EXAMPLES:

Any commutative ring is a commutative algebra over itself:

sage: A = sage.rings.ring.CommutativeAlgebra
sage: A(ZZ).is_commutative()
True
sage: A(QQ).is_commutative()
True

Trying to create a commutative algebra over a non-commutative ring will result in a TypeError.

class sage.rings.ring.CommutativeRing

Bases: sage.rings.ring.Ring

Generic commutative ring.

derivation(arg=None, twist=None)

Return the twisted or untwisted derivation over this ring specified by arg.

Note

A twisted derivation with respect to \(\theta\) (or a \(\theta\)-derivation for short) is an additive map \(d\) satisfying the following axiom for all \(x, y\) in the domain:

\[d(xy) = \theta(x) d(y) + d(x) y.\]

INPUT:

• arg – (optional) a generator or a list of coefficients that defines the derivation

• twist – (optional) the twisting homomorphism

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: R.derivation()
d/dx

In that case, arg could be a generator:

sage: R.derivation(y)
d/dy

or a list of coefficients:

sage: R.derivation([1,2,3])
d/dx + 2*d/dy + 3*d/dz

It is not possible to define derivations with respect to a polynomial which is not a variable:

sage: R.derivation(x^2)
Traceback (most recent call last):
...
ValueError: unable to create the derivation

Here is an example with twisted derivations:

sage: R.<x,y,z> = QQ[]
sage: theta = R.hom([x^2, y^2, z^2])
sage: f = R.derivation(twist=theta); f
0
sage: f.parent()
Module of twisted derivations over Multivariate Polynomial Ring in x, y, z
 over Rational Field (twisting morphism: x |--> x^2, y |--> y^2, z |--> z^2)

Specifying a scalar, the returned twisted derivation is the corresponding multiple of \(\theta - id\):

sage: R.derivation(1, twist=theta)
[x |--> x^2, y |--> y^2, z |--> z^2] - id
sage: R.derivation(x, twist=theta)
x*([x |--> x^2, y |--> y^2, z |--> z^2] - id)

derivation_module(codomain=None, twist=None)

Returns the module of derivations over this ring.

INPUT:

• codomain – an algebra over this ring or a ring homomorphism whose domain is this ring or None (default: None); if it is a morphism, the codomain of derivations will be the codomain of the morphism viewed as an algebra over self through the given morphism; if None, the codomain will be this ring

• twist – a morphism from this ring to codomain or None (default: None); if None, the coercion map from this ring to codomain will be used

Note

A twisted derivation with respect to \(\theta\) (or a \(\theta\)-derivation for short) is an additive map \(d\) satisfying the following axiom for all \(x, y\) in the domain:

\[d(xy) = \theta(x) d(y) + d(x) y.\]

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: M = R.derivation_module(); M
Module of derivations over Multivariate Polynomial Ring in x, y, z over Rational Field
sage: M.gens()
(d/dx, d/dy, d/dz)

We can specify a different codomain:

sage: K = R.fraction_field()
sage: M = R.derivation_module(K); M
Module of derivations from Multivariate Polynomial Ring in x, y, z over
 Rational Field to Fraction Field of Multivariate Polynomial Ring in x, y, z over Rational Field
sage: M.gen() / x
1/x*d/dx

Here is an example with a non-canonical defining morphism:

sage: ev = R.hom([QQ(0), QQ(1), QQ(2)])
sage: ev
Ring morphism:
  From: Multivariate Polynomial Ring in x, y, z over Rational Field
  To:   Rational Field
  Defn: x |--> 0
        y |--> 1
        z |--> 2
sage: M = R.derivation_module(ev)
sage: M
Module of derivations from Multivariate Polynomial Ring in x, y, z over Rational Field to Rational Field

Elements in \(M\) acts as derivations at \((0,1,2)\):

sage: Dx = M.gen(0); Dx
d/dx
sage: Dy = M.gen(1); Dy
d/dy
sage: Dz = M.gen(2); Dz
d/dz
sage: f = x^2 + y^2 + z^2
sage: Dx(f)  # = 2*x evaluated at (0,1,2)
0
sage: Dy(f)  # = 2*y evaluated at (0,1,2)
2
sage: Dz(f)  # = 2*z evaluated at (0,1,2)
4

An example with a twisting homomorphism:

sage: theta = R.hom([x^2, y^2, z^2])
sage: M = R.derivation_module(twist=theta); M
Module of twisted derivations over Multivariate Polynomial Ring in x, y, z
 over Rational Field (twisting morphism: x |--> x^2, y |--> y^2, z |--> z^2)

See also

derivation()

extension(poly, name=None, names=None, **kwds)

Algebraically extends self by taking the quotient self[x] / (f(x)).

INPUT:

• poly – A polynomial whose coefficients are coercible into self

• name – (optional) name for the root of \(f\)

Note

Using this method on an algebraically complete field does not return this field; the construction self[x] / (f(x)) is done anyway.

EXAMPLES:

sage: R = QQ['x']
sage: y = polygen(R)
sage: R.extension(y^2 - 5, 'a')
Univariate Quotient Polynomial Ring in a over Univariate Polynomial Ring in x over Rational Field with modulus a^2 - 5

sage: P.<x> = PolynomialRing(GF(5))
sage: F.<a> = GF(5).extension(x^2 - 2)
sage: P.<t> = F[]
sage: R.<b> = F.extension(t^2 - a); R
Univariate Quotient Polynomial Ring in b over Finite Field in a of size 5^2 with modulus b^2 + 4*a

fraction_field()

Return the fraction field of self.

EXAMPLES:

sage: R = Integers(389)['x,y']
sage: Frac(R)
Fraction Field of Multivariate Polynomial Ring in x, y over Ring of integers modulo 389
sage: R.fraction_field()
Fraction Field of Multivariate Polynomial Ring in x, y over Ring of integers modulo 389

frobenius_endomorphism(n=1)

INPUT:

• n – a nonnegative integer (default: 1)

OUTPUT:

The \(n\)-th power of the absolute arithmetic Frobenius endomorphism on this finite field.

EXAMPLES:

sage: K.<u> = PowerSeriesRing(GF(5))
sage: Frob = K.frobenius_endomorphism(); Frob
Frobenius endomorphism x |--> x^5 of Power Series Ring in u over Finite Field of size 5
sage: Frob(u)
u^5

We can specify a power:

sage: f = K.frobenius_endomorphism(2); f
Frobenius endomorphism x |--> x^(5^2) of Power Series Ring in u over Finite Field of size 5
sage: f(1+u)
1 + u^25

ideal_monoid()

Return the monoid of ideals of this ring.

EXAMPLES:

sage: ZZ.ideal_monoid()
Monoid of ideals of Integer Ring
sage: R.<x>=QQ[]; R.ideal_monoid()
Monoid of ideals of Univariate Polynomial Ring in x over Rational Field

is_commutative()

Return True, since this ring is commutative.

EXAMPLES:

sage: QQ.is_commutative()
True
sage: ZpCA(7).is_commutative()
True
sage: A = QuaternionAlgebra(QQ, -1, -3, names=('i','j','k')); A
Quaternion Algebra (-1, -3) with base ring Rational Field
sage: A.is_commutative()
False

krull_dimension()

Return the Krull dimension of this commutative ring.

The Krull dimension is the length of the longest ascending chain of prime ideals.

localization(additional_units, names=None, normalize=True, category=None)

Return the localization of self at the given additional units.

EXAMPLES:

sage: R.<x, y> = GF(3)[]
sage: R.localization((x*y, x**2+y**2))
Multivariate Polynomial Ring in x, y over Finite Field of size 3 localized at (y, x, x^2 + y^2)
sage: ~y in _
True

class sage.rings.ring.DedekindDomain

Bases: sage.rings.ring.IntegralDomain

Generic Dedekind domain class.

A Dedekind domain is a Noetherian integral domain of Krull dimension one that is integrally closed in its field of fractions.

This class is deprecated, and not actually used anywhere in the Sage code base. If you think you need it, please create a category DedekindDomains, move the code of this class there, and use it instead.

integral_closure()

Return self since Dedekind domains are integrally closed.

EXAMPLES:

sage: K = NumberField(x^2 + 1, 's')
sage: OK = K.ring_of_integers()
sage: OK.integral_closure()
Gaussian Integers in Number Field in s with defining polynomial x^2 + 1
sage: OK.integral_closure() == OK
True

sage: QQ.integral_closure() == QQ
True

is_integrally_closed()

Return True since Dedekind domains are integrally closed.

EXAMPLES:

The following are examples of Dedekind domains (Noetherian integral domains of Krull dimension one that are integrally closed over its field of fractions).

sage: ZZ.is_integrally_closed()
True
sage: K = NumberField(x^2 + 1, 's')
sage: OK = K.ring_of_integers()
sage: OK.is_integrally_closed()
True

These, however, are not Dedekind domains:

sage: QQ.is_integrally_closed()
True
sage: S = ZZ[sqrt(5)]; S.is_integrally_closed()
False
sage: T.<x,y> = PolynomialRing(QQ,2); T
Multivariate Polynomial Ring in x, y over Rational Field
sage: T.is_integral_domain()
True

is_noetherian()

Return True since Dedekind domains are Noetherian.

EXAMPLES:

The integers, \(\ZZ\), and rings of integers of number fields are Dedekind domains:

sage: ZZ.is_noetherian()
True
sage: K = NumberField(x^2 + 1, 's')
sage: OK = K.ring_of_integers()
sage: OK.is_noetherian()
True
sage: QQ.is_noetherian()
True

krull_dimension()

Return 1 since Dedekind domains have Krull dimension 1.

EXAMPLES:

The following are examples of Dedekind domains (Noetherian integral domains of Krull dimension one that are integrally closed over its field of fractions):

sage: ZZ.krull_dimension()
1
sage: K = NumberField(x^2 + 1, 's')
sage: OK = K.ring_of_integers()
sage: OK.krull_dimension()
1

The following are not Dedekind domains but have a krull_dimension function:

sage: QQ.krull_dimension()
0
sage: T.<x,y> = PolynomialRing(QQ,2); T
Multivariate Polynomial Ring in x, y over Rational Field
sage: T.krull_dimension()
2
sage: U.<x,y,z> = PolynomialRing(ZZ,3); U
Multivariate Polynomial Ring in x, y, z over Integer Ring
sage: U.krull_dimension()
4

sage: K.<i> = QuadraticField(-1)
sage: R = K.order(2*i); R
Order in Number Field in i with defining polynomial x^2 + 1 with i = 1*I
sage: R.is_maximal()
False
sage: R.krull_dimension()
1

class sage.rings.ring.EuclideanDomain

Bases: sage.rings.ring.PrincipalIdealDomain

Generic Euclidean domain class.

This class is deprecated. Please use the EuclideanDomains category instead.

parameter()

Return an element of degree 1.

EXAMPLES:

sage: R.<x>=QQ[]
sage: R.parameter()
x

class sage.rings.ring.Field

Bases: sage.rings.ring.PrincipalIdealDomain

Generic field

algebraic_closure()

Return the algebraic closure of self.

Note

This is only implemented for certain classes of field.

EXAMPLES:

sage: K = PolynomialRing(QQ,'x').fraction_field(); K
Fraction Field of Univariate Polynomial Ring in x over Rational Field
sage: K.algebraic_closure()
Traceback (most recent call last):
...
NotImplementedError: Algebraic closures of general fields not implemented.

divides(x, y, coerce=True)

Return True if x divides y in this field (usually True in a field!). If coerce is True (the default), first coerce x and y into self.

EXAMPLES:

sage: QQ.divides(2, 3/4)
True
sage: QQ.divides(0, 5)
False

fraction_field()

Return the fraction field of self.

EXAMPLES:

Since fields are their own field of fractions, we simply get the original field in return:

sage: QQ.fraction_field()
Rational Field
sage: RR.fraction_field()
Real Field with 53 bits of precision
sage: CC.fraction_field()
Complex Field with 53 bits of precision

sage: F = NumberField(x^2 + 1, 'i')
sage: F.fraction_field()
Number Field in i with defining polynomial x^2 + 1

ideal(*gens, **kwds)

Return the ideal generated by gens.

EXAMPLES:

sage: QQ.ideal(2)
Principal ideal (1) of Rational Field
sage: QQ.ideal(0)
Principal ideal (0) of Rational Field

integral_closure()

Return this field, since fields are integrally closed in their fraction field.

EXAMPLES:

sage: QQ.integral_closure()
Rational Field
sage: Frac(ZZ['x,y']).integral_closure()
Fraction Field of Multivariate Polynomial Ring in x, y over Integer Ring

is_field(proof=True)

Return True since this is a field.

EXAMPLES:

sage: Frac(ZZ['x,y']).is_field()
True

is_integrally_closed()

Return True since fields are trivially integrally closed in their fraction field (since they are their own fraction field).

EXAMPLES:

sage: Frac(ZZ['x,y']).is_integrally_closed()
True

is_noetherian()

Return True since fields are Noetherian rings.

EXAMPLES:

sage: QQ.is_noetherian()
True

krull_dimension()

Return the Krull dimension of this field, which is 0.

EXAMPLES:

sage: QQ.krull_dimension()
0
sage: Frac(QQ['x,y']).krull_dimension()
0

prime_subfield()

Return the prime subfield of self.

EXAMPLES:

sage: k = GF(9, 'a')
sage: k.prime_subfield()
Finite Field of size 3

class sage.rings.ring.IntegralDomain

Bases: sage.rings.ring.CommutativeRing

Generic integral domain class.

This class is deprecated. Please use the sage.categories.integral_domains.IntegralDomains category instead.

is_field(proof=True)

Return True if this ring is a field.

EXAMPLES:

sage: GF(7).is_field()
True

The following examples have their own is_field implementations:

sage: ZZ.is_field(); QQ.is_field()
False
True
sage: R.<x> = PolynomialRing(QQ); R.is_field()
False

is_integral_domain(proof=True)

Return True, since this ring is an integral domain.

(This is a naive implementation for objects with type IntegralDomain)

EXAMPLES:

sage: ZZ.is_integral_domain()
True
sage: QQ.is_integral_domain()
True
sage: ZZ['x'].is_integral_domain()
True
sage: R = ZZ.quotient(ZZ.ideal(10)); R.is_integral_domain()
False

is_integrally_closed()

Return True if this ring is integrally closed in its field of fractions; otherwise return False.

When no algorithm is implemented for this, then this function raises a NotImplementedError.

Note that is_integrally_closed has a naive implementation in fields. For every field \(F\), \(F\) is its own field of fractions, hence every element of \(F\) is integral over \(F\).

EXAMPLES:

sage: ZZ.is_integrally_closed()
True
sage: QQ.is_integrally_closed()
True
sage: QQbar.is_integrally_closed()
True
sage: GF(5).is_integrally_closed()
True
sage: Z5 = Integers(5); Z5
Ring of integers modulo 5
sage: Z5.is_integrally_closed()
Traceback (most recent call last):
...
AttributeError: 'IntegerModRing_generic_with_category' object has no attribute 'is_integrally_closed'

class sage.rings.ring.NoetherianRing

Bases: sage.rings.ring.CommutativeRing

Generic Noetherian ring class.

A Noetherian ring is a commutative ring in which every ideal is finitely generated.

This class is deprecated, and not actually used anywhere in the Sage code base. If you think you need it, please create a category NoetherianRings, move the code of this class there, and use it instead.

is_noetherian()

Return True since this ring is Noetherian.

EXAMPLES:

sage: ZZ.is_noetherian()
True
sage: QQ.is_noetherian()
True
sage: R.<x> = PolynomialRing(QQ)
sage: R.is_noetherian()
True

class sage.rings.ring.PrincipalIdealDomain

Bases: sage.rings.ring.IntegralDomain

Generic principal ideal domain.

This class is deprecated. Please use the PrincipalIdealDomains category instead.

class_group()

Return the trivial group, since the class group of a PID is trivial.

EXAMPLES:

sage: QQ.class_group()
Trivial Abelian group

content(x, y, coerce=True)

Return the content of \(x\) and \(y\), i.e. the unique element \(c\) of self such that \(x/c\) and \(y/c\) are coprime and integral.

EXAMPLES:

sage: QQ.content(ZZ(42), ZZ(48)); type(QQ.content(ZZ(42), ZZ(48)))
6
<class 'sage.rings.rational.Rational'>
sage: QQ.content(1/2, 1/3)
1/6
sage: factor(1/2); factor(1/3); factor(1/6)
2^-1
3^-1
2^-1 * 3^-1
sage: a = (2*3)/(7*11); b = (13*17)/(19*23)
sage: factor(a); factor(b); factor(QQ.content(a,b))
2 * 3 * 7^-1 * 11^-1
13 * 17 * 19^-1 * 23^-1
7^-1 * 11^-1 * 19^-1 * 23^-1

Note the changes to the second entry:

sage: c = (2*3)/(7*11); d = (13*17)/(7*19*23)
sage: factor(c); factor(d); factor(QQ.content(c,d))
2 * 3 * 7^-1 * 11^-1
7^-1 * 13 * 17 * 19^-1 * 23^-1
7^-1 * 11^-1 * 19^-1 * 23^-1
sage: e = (2*3)/(7*11); f = (13*17)/(7^3*19*23)
sage: factor(e); factor(f); factor(QQ.content(e,f))
2 * 3 * 7^-1 * 11^-1
7^-3 * 13 * 17 * 19^-1 * 23^-1
7^-3 * 11^-1 * 19^-1 * 23^-1

gcd(x, y, coerce=True)

Return the greatest common divisor of x and y, as elements of self.

EXAMPLES:

The integers are a principal ideal domain and hence a GCD domain:

sage: ZZ.gcd(42, 48)
6
sage: 42.factor(); 48.factor()
2 * 3 * 7
2^4 * 3
sage: ZZ.gcd(2^4*7^2*11, 2^3*11*13)
88
sage: 88.factor()
2^3 * 11

In a field, any nonzero element is a GCD of any nonempty set of nonzero elements. In previous versions, Sage used to return 1 in the case of the rational field. However, since trac ticket #10771, the rational field is considered as the fraction field of the integer ring. For the fraction field of an integral domain that provides both GCD and LCM, it is possible to pick a GCD that is compatible with the GCD of the base ring:

sage: QQ.gcd(ZZ(42), ZZ(48)); type(QQ.gcd(ZZ(42), ZZ(48)))
6
<class 'sage.rings.rational.Rational'>
sage: QQ.gcd(1/2, 1/3)
1/6

Polynomial rings over fields are GCD domains as well. Here is a simple example over the ring of polynomials over the rationals as well as over an extension ring. Note that gcd requires x and y to be coercible:

sage: R.<x> = PolynomialRing(QQ)
sage: S.<a> = NumberField(x^2 - 2, 'a')
sage: f = (x - a)*(x + a); g = (x - a)*(x^2 - 2)
sage: print(f); print(g)
x^2 - 2
x^3 - a*x^2 - 2*x + 2*a
sage: f in R
True
sage: g in R
False
sage: R.gcd(f,g)
Traceback (most recent call last):
...
TypeError: Unable to coerce 2*a to a rational
sage: R.base_extend(S).gcd(f,g)
x^2 - 2
sage: R.base_extend(S).gcd(f, (x - a)*(x^2 - 3))
x - a

is_noetherian()

Every principal ideal domain is noetherian, so we return True.

EXAMPLES:

sage: Zp(5).is_noetherian()
True

class sage.rings.ring.Ring

Bases: sage.structure.parent_gens.ParentWithGens

Generic ring class.

base_extend(R)

EXAMPLES:

sage: QQ.base_extend(GF(7))
Traceback (most recent call last):
...
TypeError: no base extension defined
sage: ZZ.base_extend(GF(7))
Finite Field of size 7

category()

Return the category to which this ring belongs.

Note

This method exists because sometimes a ring is its own base ring. During initialisation of a ring \(R\), it may be checked whether the base ring (hence, the ring itself) is a ring. Hence, it is necessary that R.category() tells that R is a ring, even before its category is properly initialised.

EXAMPLES:

sage: FreeAlgebra(QQ, 3, 'x').category() # todo: use a ring which is not an algebra!
Category of algebras with basis over Rational Field

Since a quotient of the integers is its own base ring, and during initialisation of a ring it is tested whether the base ring belongs to the category of rings, the following is an indirect test that the category() method of rings returns the category of rings even before the initialisation was successful:

sage: I = Integers(15)
sage: I.base_ring() is I
True
sage: I.category()
Join of Category of finite commutative rings
    and Category of subquotients of monoids
    and Category of quotients of semigroups
    and Category of finite enumerated sets

epsilon()

Return the precision error of elements in this ring.

EXAMPLES:

sage: RDF.epsilon()
2.220446049250313e-16
sage: ComplexField(53).epsilon()
2.22044604925031e-16
sage: RealField(10).epsilon()
0.0020

For exact rings, zero is returned:

sage: ZZ.epsilon()
0

This also works over derived rings:

sage: RR['x'].epsilon()
2.22044604925031e-16
sage: QQ['x'].epsilon()
0

For the symbolic ring, there is no reasonable answer:

sage: SR.epsilon()
Traceback (most recent call last):
...
NotImplementedError

ideal(*args, **kwds)

Return the ideal defined by x, i.e., generated by x.

INPUT:

• *x – list or tuple of generators (or several input arguments)

• coerce – bool (default: True); this must be a keyword argument. Only set it to False if you are certain that each generator is already in the ring.

• ideal_class – callable (default: self._ideal_class_()); this must be a keyword argument. A constructor for ideals, taking the ring as the first argument and then the generators. Usually a subclass of Ideal_generic or Ideal_nc.

• Further named arguments (such as side in the case of non-commutative rings) are forwarded to the ideal class.

EXAMPLES:

sage: R.<x,y> = QQ[]
sage: R.ideal(x,y)
Ideal (x, y) of Multivariate Polynomial Ring in x, y over Rational Field
sage: R.ideal(x+y^2)
Ideal (y^2 + x) of Multivariate Polynomial Ring in x, y over Rational Field
sage: R.ideal( [x^3,y^3+x^3] )
Ideal (x^3, x^3 + y^3) of Multivariate Polynomial Ring in x, y over Rational Field

Here is an example over a non-commutative ring:

sage: A = SteenrodAlgebra(2)
sage: A.ideal(A.1,A.2^2)
Twosided Ideal (Sq(2), Sq(2,2)) of mod 2 Steenrod algebra, milnor basis
sage: A.ideal(A.1,A.2^2,side='left')
Left Ideal (Sq(2), Sq(2,2)) of mod 2 Steenrod algebra, milnor basis

ideal_monoid()

Return the monoid of ideals of this ring.

EXAMPLES:

sage: F.<x,y,z> = FreeAlgebra(ZZ, 3)
sage: I = F*[x*y+y*z,x^2+x*y-y*x-y^2]*F
sage: Q = F.quotient(I)
sage: Q.ideal_monoid()
Monoid of ideals of Quotient of Free Algebra on 3 generators (x, y, z) over Integer Ring by the ideal (x*y + y*z, x^2 + x*y - y*x - y^2)
sage: F.<x,y,z> = FreeAlgebra(ZZ, implementation='letterplace')
sage: I = F*[x*y+y*z,x^2+x*y-y*x-y^2]*F
sage: Q = F.quo(I)
sage: Q.ideal_monoid()
Monoid of ideals of Quotient of Free Associative Unital Algebra on 3 generators (x, y, z) over Integer Ring by the ideal (x*y + y*z, x*x + x*y - y*x - y*y)

is_commutative()

Return True if this ring is commutative.

EXAMPLES:

sage: QQ.is_commutative()
True
sage: QQ['x,y,z'].is_commutative()
True
sage: Q.<i,j,k> = QuaternionAlgebra(QQ, -1,-1)
sage: Q.is_commutative()
False

is_exact()

Return True if elements of this ring are represented exactly, i.e., there is no precision loss when doing arithmetic.

Note

This defaults to True, so even if it does return True you have no guarantee (unless the ring has properly overloaded this).

EXAMPLES:

sage: QQ.is_exact()    # indirect doctest
True
sage: ZZ.is_exact()
True
sage: Qp(7).is_exact()
False
sage: Zp(7, type='capped-abs').is_exact()
False

is_field(proof=True)

Return True if this ring is a field.

INPUT:

• proof – (default: True) Determines what to do in unknown cases

ALGORITHM:

If the parameter proof is set to True, the returned value is correct but the method might throw an error. Otherwise, if it is set to False, the method returns True if it can establish that self is a field and False otherwise.

EXAMPLES:

sage: QQ.is_field()
True
sage: GF(9,'a').is_field()
True
sage: ZZ.is_field()
False
sage: QQ['x'].is_field()
False
sage: Frac(QQ['x']).is_field()
True

This illustrates the use of the proof parameter:

sage: R.<a,b> = QQ[]
sage: S.<x,y> = R.quo((b^3))
sage: S.is_field(proof = True)
Traceback (most recent call last):
...
NotImplementedError
sage: S.is_field(proof = False)
False

is_integral_domain(proof=True)

Return True if this ring is an integral domain.

INPUT:

• proof – (default: True) Determines what to do in unknown cases

ALGORITHM:

If the parameter proof is set to True, the returned value is correct but the method might throw an error. Otherwise, if it is set to False, the method returns True if it can establish that self is an integral domain and False otherwise.

EXAMPLES:

sage: QQ.is_integral_domain()
True
sage: ZZ.is_integral_domain()
True
sage: ZZ['x,y,z'].is_integral_domain()
True
sage: Integers(8).is_integral_domain()
False
sage: Zp(7).is_integral_domain()
True
sage: Qp(7).is_integral_domain()
True
sage: R.<a,b> = QQ[]
sage: S.<x,y> = R.quo((b^3))
sage: S.is_integral_domain()
False

This illustrates the use of the proof parameter:

sage: R.<a,b> = ZZ[]
sage: S.<x,y> = R.quo((b^3))
sage: S.is_integral_domain(proof = True)
Traceback (most recent call last):
...
NotImplementedError
sage: S.is_integral_domain(proof = False)
False

is_noetherian()

Return True if this ring is Noetherian.

EXAMPLES:

sage: QQ.is_noetherian()
True
sage: ZZ.is_noetherian()
True

is_prime_field()

Return True if this ring is one of the prime fields \(\QQ\) or \(\GF{p}\).

EXAMPLES:

sage: QQ.is_prime_field()
True
sage: GF(3).is_prime_field()
True
sage: GF(9,'a').is_prime_field()
False
sage: ZZ.is_prime_field()
False
sage: QQ['x'].is_prime_field()
False
sage: Qp(19).is_prime_field()
False

is_subring(other)

Return True if the canonical map from self to other is injective.

Raises a NotImplementedError if not known.

EXAMPLES:

sage: ZZ.is_subring(QQ)
True
sage: ZZ.is_subring(GF(19))
False

one()

Return the one element of this ring (cached), if it exists.

EXAMPLES:

sage: ZZ.one()
1
sage: QQ.one()
1
sage: QQ['x'].one()
1

The result is cached:

sage: ZZ.one() is ZZ.one()
True

order()

The number of elements of self.

EXAMPLES:

sage: GF(19).order()
19
sage: QQ.order()
+Infinity

principal_ideal(gen, coerce=True)

Return the principal ideal generated by gen.

EXAMPLES:

sage: R.<x,y> = ZZ[]
sage: R.principal_ideal(x+2*y)
Ideal (x + 2*y) of Multivariate Polynomial Ring in x, y over Integer Ring

random_element(bound=2)

Return a random integer coerced into this ring, where the integer is chosen uniformly from the interval [-bound,bound].

INPUT:

• bound – integer (default: 2)

ALGORITHM:

Uses Python’s randint.

unit_ideal()

Return the unit ideal of this ring.

EXAMPLES:

sage: Zp(7).unit_ideal()
Principal ideal (1 + O(7^20)) of 7-adic Ring with capped relative precision 20

zero()

Return the zero element of this ring (cached).

EXAMPLES:

sage: ZZ.zero()
0
sage: QQ.zero()
0
sage: QQ['x'].zero()
0

The result is cached:

sage: ZZ.zero() is ZZ.zero()
True

zero_ideal()

Return the zero ideal of this ring (cached).

EXAMPLES:

sage: ZZ.zero_ideal()
Principal ideal (0) of Integer Ring
sage: QQ.zero_ideal()
Principal ideal (0) of Rational Field
sage: QQ['x'].zero_ideal()
Principal ideal (0) of Univariate Polynomial Ring in x over Rational Field

The result is cached:

sage: ZZ.zero_ideal() is ZZ.zero_ideal()
True

zeta(n=2, all=False)

Return a primitive n-th root of unity in self if there is one, or raise a ValueError otherwise.

INPUT:

• n – positive integer

• all – bool (default: False) - whether to return a list of all primitive \(n\)-th roots of unity. If True, raise a ValueError if self is not an integral domain.

OUTPUT:

Element of self of finite order

EXAMPLES:

sage: QQ.zeta()
-1
sage: QQ.zeta(1)
1
sage: CyclotomicField(6).zeta(6)
zeta6
sage: CyclotomicField(3).zeta(3)
zeta3
sage: CyclotomicField(3).zeta(3).multiplicative_order()
3
sage: a = GF(7).zeta(); a
3
sage: a.multiplicative_order()
6
sage: a = GF(49,'z').zeta(); a
z
sage: a.multiplicative_order()
48
sage: a = GF(49,'z').zeta(2); a
6
sage: a.multiplicative_order()
2
sage: QQ.zeta(3)
Traceback (most recent call last):
...
ValueError: no n-th root of unity in rational field
sage: Zp(7, prec=8).zeta()
3 + 4*7 + 6*7^2 + 3*7^3 + 2*7^5 + 6*7^6 + 2*7^7 + O(7^8)

zeta_order()

Return the order of the distinguished root of unity in self.

EXAMPLES:

sage: CyclotomicField(19).zeta_order()
38
sage: GF(19).zeta_order()
18
sage: GF(5^3,'a').zeta_order()
124
sage: Zp(7, prec=8).zeta_order()
6

sage.rings.ring.is_Ring(x)

Return True if x is a ring.

EXAMPLES:

sage: from sage.rings.ring import is_Ring
sage: is_Ring(ZZ)
True
sage: MS = MatrixSpace(QQ,2)
sage: is_Ring(MS)
True