Division rings

class sage.categories.division_rings.DivisionRings(base_category)

Bases: CategoryWithAxiom_singleton

The category of division rings.

A division ring (or skew field) is a not necessarily commutative ring where all nonzero elements have multiplicative inverses

EXAMPLES:

Sage

sage: DivisionRings()
Category of division rings
sage: DivisionRings().super_categories()
[Category of domains]

Python

>>> from sage.all import *
>>> DivisionRings()
Category of division rings
>>> DivisionRings().super_categories()
[Category of domains]

Commutative

alias of Fields

class ElementMethods

Bases: object

Finite_extra_super_categories()

Return extraneous super categories for DivisionRings().Finite().

EXAMPLES:

Any field is a division ring:

Sage

sage: Fields().is_subcategory(DivisionRings())
True

Python

>>> from sage.all import *
>>> Fields().is_subcategory(DivisionRings())
True

This methods specifies that, by Weddeburn theorem, the reciprocal holds in the finite case: a finite division ring is commutative and thus a field:

Sage

sage: DivisionRings().Finite_extra_super_categories()
(Category of commutative magmas,)
sage: DivisionRings().Finite()
Category of finite enumerated fields

Python

>>> from sage.all import *
>>> DivisionRings().Finite_extra_super_categories()
(Category of commutative magmas,)
>>> DivisionRings().Finite()
Category of finite enumerated fields

Warning

This is not implemented in DivisionRings.Finite.extra_super_categories because the categories of finite division rings and of finite fields coincide. See the section Deduction rules in the documentation of axioms.

class ParentMethods

Bases: object

extra_super_categories()

Return the Domains category.

This method specifies that a division ring has no zero divisors, i.e. is a domain.

See also

The Deduction rules section in the documentation of axioms

EXAMPLES:

Sage

sage: DivisionRings().extra_super_categories()
(Category of domains,)
sage: "NoZeroDivisors" in DivisionRings().axioms()
True

Python

>>> from sage.all import *
>>> DivisionRings().extra_super_categories()
(Category of domains,)
>>> "NoZeroDivisors" in DivisionRings().axioms()
True