Division rings

class sage.categories.division_rings.DivisionRings(base_category)

Bases: sage.categories.category_with_axiom.CategoryWithAxiom_singleton

The category of division rings

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


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

alias of Fields

class ElementMethods

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


Any field is a division ring:

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

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


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

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


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