Division rings¶
- class sage.categories.division_rings.DivisionRings(base_category)[source]¶
- 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: DivisionRings() Category of division rings sage: DivisionRings().super_categories() [Category of domains] - >>> from sage.all import * >>> DivisionRings() Category of division rings >>> DivisionRings().super_categories() [Category of domains] - Finite_extra_super_categories()[source]¶
- Return extraneous super categories for - DivisionRings().Finite().- EXAMPLES: - Any field is a division ring: - sage: Fields().is_subcategory(DivisionRings()) True - >>> 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: DivisionRings().Finite_extra_super_categories() (Category of commutative magmas,) sage: DivisionRings().Finite() Category of finite enumerated fields - >>> 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_categoriesbecause the categories of finite division rings and of finite fields coincide. See the section Deduction rules in the documentation of axioms.
 - extra_super_categories()[source]¶
- Return the - Domainscategory.- 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: DivisionRings().extra_super_categories() (Category of domains,) sage: "NoZeroDivisors" in DivisionRings().axioms() True - >>> from sage.all import * >>> DivisionRings().extra_super_categories() (Category of domains,) >>> "NoZeroDivisors" in DivisionRings().axioms() True