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 Field, 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 (to be deprecated)

  • Algebra (deprecated and essentially removed)

  • CommutativeRing (deprecated and essentially removed)

    • NoetherianRing (deprecated and essentially removed)

    • CommutativeAlgebra (deprecated and essentially removed)

    • IntegralDomain (deprecated and essentially removed)

      • DedekindDomain (deprecated and essentially removed)

      • PrincipalIdealDomain (deprecated and essentially removed)

Other subclaasses of Ring are

• 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.

(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 deprecated 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: Ring

class sage.rings.ring.CommutativeAlgebra

Bases: Ring

class sage.rings.ring.CommutativeRing

Bases: Ring

class sage.rings.ring.DedekindDomain

Bases: Ring

class sage.rings.ring.Field

Bases: Ring

Generic field

class sage.rings.ring.IntegralDomain

Bases: Ring

class sage.rings.ring.NoetherianRing

Bases: Ring

class sage.rings.ring.PrincipalIdealDomain

Bases: Ring

class sage.rings.ring.Ring

Bases: Parent

Generic ring class.

base_extend(X)

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> QQ.base_extend(GF(Integer(7)))
Traceback (most recent call last):
...
TypeError: no base extension defined
>>> ZZ.base_extend(GF(Integer(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

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

Python

>>> from sage.all import *
>>> FreeAlgebra(QQ, Integer(3), 'x').category()  # todo: use a ring which is not an algebra!   # needs sage.combinat sage.modules
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

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

[Python]

>>> from sage.all import *
>>> I = Integers(Integer(15))
>>> I.base_ring() is I
True
>>> 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

gens()

Return a tuple whose entries are the generators for this object, in order.

EXAMPLES:

Sage

sage: ZZ.gens()
(1,)

sage: ZZ["y"].gens()
(y,)

sage: InfinitePolynomialRing(QQ, "a").gens()
(a_*,)

Python

>>> from sage.all import *
>>> ZZ.gens()
(1,)

>>> ZZ["y"].gens()
(y,)

>>> InfinitePolynomialRing(QQ, "a").gens()
(a_*,)

one()

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

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> ZZ.one()
1
>>> QQ.one()
1
>>> QQ['x'].one()
1

The result is cached:

Sage

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

[Python]

>>> from sage.all import *
>>> ZZ.one() is ZZ.one()
True

order()

The number of elements of self.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> GF(Integer(19)).order()
19
>>> QQ.order()
+Infinity

zero()

Return the zero element of this ring (cached).

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> ZZ.zero()
0
>>> QQ.zero()
0
>>> QQ['x'].zero()
0

The result is cached:

Sage

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

[Python]

>>> from sage.all import *
>>> ZZ.zero() is ZZ.zero()
True